Student Project Abstracts 2005 - Pluto - University of Washington

EDWARDS1Asymmetric System Coupled to 25 Oscillators1e+06Power Spectrum of 25 Oscillators Coupled to a Spin System0.810000010.80.60.40.20-0.2-0.4-0.60.610000Asymmetric System Coupled to 25 Oscillators0.40.20-0.2-0.4-0.6-0.80.010 50 100 150 200-0.80 50 100 150 200ttP (υ)1e+061000001000010001001010.1Power Spectrum of 25 Oscillators Coupled to a Spin SystemP (υ)10001001010.10 0.05 0.1 0.15 0.2 0.25υ0.010 0.05 0.1 0.15 0.2 0.25υigure 3: An AsymmetricFigure System 3. An Coupled Asymmetric to 25System Oscillators. Coupled The to 25 frequencies Oscillators. The of the frequencies bath oscillators, of the bath oscillators, ω, are evenly spaced between 3”, are evenly spaced between 3ε 5εto . Using the same Ω, Ω, ε, and ε, and coupling coupling as Figs. as1 Figs. and 2, 1the and spin 2, system the spin decays in the presence of more oscillators.2 2ystem Figure decays in 3: the An presence Asymmetric of more oscillators. System Coupled to 25 Oscillators. The frequencies of the bath oscillators,ω, are evenly spaced between 3ε 5ε2to2Using the same Ω, ε, and coupling as Figs. 1 and 2, the spinAdding single bath oscillatorsCONCLUSIONfrequencies, that are not close the spin system’s naturalsystem decays in the presence of more oscillators.ACKNOWLEDGEMENTSrequency, cause relatively simple behavior in the system. This behavior of σ z coupled to a singleath oscillatorThe off of model ω 0 canhere be described has proven as to energy be non-trivial passing from with theinterestingspin system into theResearch harmonic support is gratefully acknowledged from the Na-is theScience flow of Foundation Center on Materials and Devices forscillator behavior. and then returning The approximations into the spinmade system. in QHD The lower seem frequency to give reasonablyAdding and forthoscillationtionalnergy backphysical single fromresultsspinbath towhileboson. oscillators suitably simplifying at frequencies, the dynamics. that Information are not Technology close Research to the (CMDITR), spin system’s DMR-0120967. naturalHowever, for single oscillators where ωfrequency, However, there cause is still relatively much to j is on ωexplore simple0 cause more complicated behavior. In Figure 2and verify with this model.oscillator The with frequency there this ≈ off ω o shows gives a responses over a range of frequencies and of greaterhe oscillator at ω j =behavior in the system. This behavior of σ z coupled to a single9ε4re and bath However, verify agnitude. This behavior needs model. still the of tomuch new be ω The spinoscillation tofrequency explore and induced of the verify by new coupling0studied can moredescribed depth. aswith energy spinharmonicis model. oscillation a oscillator harmonic induced oscillator should by should be coupling determined be to a harmonic as as function a oscillator of ω j should be determined as a function of ωthis model. passingThe from frequency the spin of the system new spin-into the harmonicoscillator The and frequency thenofreturning new spinshouldesponse . Larger and and be from perhaps determined Baths adding g j . oscillators Also, as athe the function frequency near ωof 0 response ωshould jinto the spin system. The lower frequency oscillation jis the flow ofresponse from be determined. from adding adding oscillatorsdifference In realistic addition near condensed ωbetween 0 should to frequency phase be treating bedetermined.involves responses, the many system bath the oscillators. asdifference closed The with between last plot treating Figure 2the system as closed withoscillators near ω 0 should be determined.energy back and forth from spin to boson.es, oscillators Athe morehowstween frequencies the a treating finite However, power In or set addition spectrum as the ofan for system oscillators open offrequency single multiple system, as closed bath responses, discrete oscillators, like with a the frequencies Langevin where coupled difference to ωequation,j or the between isas spin ansystem, ωopen 0 cause with system, all more thelikecomplicated a Langevin equation, behavior. In Figure 2hownntral open densityfrequencieswhere treating system, the shouldexceptthe like bathfor system abe hasωLangevin addressed.=the oscillator closed continuous 9εj4. The= 9ε powerwith equation, a There finite spectral set could of density oscillators be a way should at dis-Langevin-type frequencies There the This QHD or could formalism, behavior as an approximation open a way making system, needs to like use intoa be of a the Langevin-type studied tools equa-of moreformalism, in depth. making use of the tools ofto be addressed. There could be a way to4 ≈ ω spectrum is more complex than a simple sum ofo shows gives a responses over a range of frequencies and of greaterhe single oscillator power spectra – most notably by the dip in the spectra just below frequenciesfbe intoν = magnitude. addressed.0.1.incorporate acretepeCoupling formalism, statistical tion, many where making bath mechanics. the oscillators bath use has ofalso continuous thechanges toolsspectral the of behavior density of σ should z in the be time domain into ael oresuccessfully physical addressed. Inresult. all, gives the There Finally, QHD could a isphysical the spin-boson be casea way with solution to many incorporate model oscillators. for successfully the QHD asymmetricFigure approximationLarger coupled3 gives shows the a physical case wheresolution the for the asymmetricpinstively aB. systemphysical system.isthe same solution into Thetoas a decay25Langevin-type the Baths oscillators,for data of the spinspacedpresented asymmetric formalism, qualitativelyevenly betweenbymaking Makri. use the 3 There 3ε 5εof 2same andthe tools 2 are as. Thethebehaviordata presentedof thisby Makri. 3 There areystem is obviously more complex thandel the to data still explore, of presented statistical many such more mechanics. byasaspects Makri. symmetric 3 just the addition of a new oscillation, as evident in theower spectra of σ z (Figure 3). It decaysofThere thistosystems,a lowermodel are energyand tostate explore, asymmetricover timesuchandas thesymmetric spin stays in asystems, and asymmetricphering owasenergy symmetric systems Athe state more In response until all, with systems, the realistic Ω numerical QHD of > εthese and as spin-boson solution condensed well asymmetric systems eventually model deciphering to phase various successfully grows unstable. involves the baths response gives Figure withamany 3of greatly these bath resembles systems oscillators. various The baths last with plot in Figure 2hee ofsame shows these different physical systemsystems the that spectral solution power is presentedto various densities. for spectrum the in Makri’s asymmetric bathssolution, of with multiple system. 3 although The batha decay moreoscillators, careful of comparison coupled still to the spin system, with all theeeds to be made.shown spin is frequencies qualitatively the except same as for the ω data = presented 9ε by Makri.4. The power 3spectrum is more complex than a simple sum of. the ConclusionTheresingleareoscillatorstill many morepoweraspectsspectraof this–modelmosttonotablyexplore,by the dip in the spectra just below frequenciessuch as symmetric systems, and asymmetric systems withof ν = 0.1.The model Ω > ε here as well has proven as deciphering to be non-trivial the response with interesting of these systems behavior. to The approximationsade in QHDvarious Coupling seem to givebaths with many reasonablydifferent bathphysical spectral oscillators results whiledensities. also suitably changes simplifying the thebehavior dynamics. of σ z in the time domain into amore physical result. Finally, is the case with many oscillators. Figure 3 shows the case where the6spin system is coupled REFERENCES to 25 oscillators, spaced evenly between 3ε 5ε2and2. The behavior of thissystem 1O. V. is Prezhdo obviously and Y. V. more Pereverzev, complex J. Chem. than Phys. just 113, 6557 the addition of a new oscillation, as evident in thepower (2000). spectra of σ z (Figure 3). It decays to a lower energy state over time and the spin stays in alow2A. energy J. Leggett state et al., until Rev. Mod. the Phys. numerical 59, 1 (1987). solution eventually grows unstable. Figure 3 greatly resembles3N. Makri, J. Math. Phys. 36, 2430 (1995).the same system that is presented in Makri’s solution, 3 although a more careful comparison stillneeds to be made.V. ConclusionCMDITR Review of Undergraduate Research Vol. 2 No. 1 Summer 2005 43