Student Project Abstracts 2005 - Pluto - University of Washington

QUANTIZED HAMILTON DYNAMICS APPLIED TO CONDENSED PHASE SPIN-RELAXATION100 Asymmetric System Coupled to 25 OscillatorsPower Spectrum of 25 Oscillators Coupled to a Spin System11e+0610010 Asymmetric System Coupled to 25 OscillatorsPower Spectrum of 25 Oscillators Coupled to a Spin System0.811000001e+06100.6 10.8100001000000.410.6 0.11000100000.20.40.011000.10Asymmetric System Coupled to 25 Oscillators1000Power Spectrum of 25 Oscillators Coupled to a Spin System0.21101e+06-0.2 0.0011000.010 0 0.050.80.1 0.15 0.2 0.250 0.05 0.1 0.15 0.2 0.251100000-0.4υ10υ-0.20.6-0.60.1100001-0.40.4-0.80.011000-0.60.10 50 0.2 100 150 2000 0.05 0.1 0.15 0.2 0.25Asymmetric System tCoupled to 25 OscillatorsPower Spectrum 100of υ25 Oscillators Coupled to a Spin System-0.8100.010 50 100 150 2001e+06 0 0.05 0.1 0.15 0.2 0.25Asymmetric System Coupled to 25 Oscillators Coupled Oscillators with ω > 9ε/4tPower Spectrum of 25 Oscillators Coupled to 10 a Spin υ System Coupled to Multiple Oscillators0.8-0.21e+061e+06100000ω = 19ε/81e+0610.6-0.4ω = 10ε/4100000Asymmetric System Coupled to 25 Oscillators 10000010000 Power 100000 Spectrum of 25 Oscillators Coupled to a Spin System10.4-0.60.11e+065ε100001000Figure 3: 100000.8 An 0.2 Asymmetric -0.8 System 2to Coupled 2Usingto the 25same Oscillators. Ω, ε, and The coupling frequencies as0.01Figs. of the 1 and bath2, oscillators, the spin0 1000001000100ω, are evenly 0.60 spaced between 3ε1000050 100 150 2000 0.05 0.1 0.15 0.2 0.255ε10002to2Usingtthe same Ω, ε, and coupling as Figs. 1 and 2, theυspin10000100010010system decays100 0.4-0.2 in the presence of more oscillators.1000100P (υ)Figure 3: An Asymmetric System Coupled to 25 Oscillators. The frequencies of the bath oscillators,ω, are evenly spaced between 3εsystem decays in the presence of more oscillators.P (υ)P (υ)P (υ)-0.41010.2Adding 10 single bath oscillators at frequencies, that are not close to the spin system’s natural100-0.6 010.1frequency, cause relatively Figure 3: simple An Asymmetric behavior in System the system. Coupled This to 25 behavior Oscillators. of σ10zThe coupled frequencies to a single of the bath oscillators,Adding single bath oscillators at frequencies, that are not close to the spin system’s natural-0.2-0.80.10.01bath oscillator 0 offω, are ω 0 50 can evenly be described spaced between 100 150 as energy 3ε1015ε2000 0.05 0.1 0.15 0.2 0.25frequency, cause relatively simple behavior in the 2tosystem. passing 2Using fromthe the same spinΩ, system ε, andinto coupling the harmonic as Figs. 1 and 2, the spinThis 1 behavior 1 of σ-0.4ttz coupled to a single0.10.01υoscillator 0 50 and then system100 returning decays150 into in the the200 spin presence system. of more0 Theoscillators.bath oscillator off of ω 0.05 0.1 0.15 0.2 0.25-0.60 can be described as energy passing lower fromfrequency 0.1the spinoscillation system intoisthe theharmonicflow of0.01tυ 0.1energy oscillator backand andthen forthreturning from spininto boson. the spin system. The lower frequency oscillation is the flow of-0.80.010 50 100 150 2000 0.05 0.1 0.15 0.2 0.25However,0.001for single oscillators where ω Figure 3: An Asymmetric SystemtCoupled j is on ω to 25 0 cause more complicated0.01energy back behavior. In Figure 20 and forth 0.05 Adding from single spin Oscillators. The frequenciesυof the bath oscillators,the oscillator at ω ω, are evenly j = 9εspaced4between 3ε 5εAn Asymmetric System Coupled ≈ ω0.1 tobath boson. 0.15 oscillators 0.2 at frequencies, 0.25 that 0 are not 0.05 close to 0.1 the spin0.15 system’s0.2 natural0.25However, forfrequency, single oscillators cause o shows relatively υtowhere gives 225 toOscillators. ω 2 Using jasimple isresponses the ωbehavior The 0 same cause overfrequencies Ω, more in a range the ε, and complicated system. of frequencies Thisof coupling the bath as behavior. and oscillators,Figs. 1 and InofgreaterFigure συ2, z coupledthe 2 to a singlespinmagnitude.nlythe spaced system oscillator Thisbetween decays atbehavior bath ω in j 3ε = oscillator the presence of more oscillators.2to 9ε needs to be studied more in depth.4 5ε ≈ 2Using ω ooff shows of ωthe samegives 0 canΩ,abe responses describedε, and couplingover as energy aasrange passingFigs.of1frequencies from theand 2, the spinand spinof system greater into the harmonicFigure 2. Power Spectra of Asymmetric Systems Coupled to Single Oscillators, using the same conditions as Figure 1. The power spectraecaysmagnitude. Figure in the 3: presence An This Asymmetric oscillator of behavior the asymmetric andof moreneeds System systems thenoscillators.to show Coupled be returning studied two major tointo oscillations. more 25 Oscillators. the in spin depth. A high system. frequency The frequencies oscillation, The lower dependent offrequency the on bath Ω and ε oscillators, oscillation and the slower oscillation is the flow ofB. ω, Larger are evenly Baths spaced energy is dependentbetween back and on ω. 3ε forth For an 5ε2to from oscillator2Using spin wheretheto ω issame boson. the natural frequency of the spin system more complicated behavior occurs.Ω, ε, and coupling as Figs. 1 and 2, the spinAdding single bath oscillators at frequencies, that are not close to the spin system’s naturalB. system Larger decays Baths in the However, presence for of single more oscillators. where ω j is on ω 0 cause more complicated behavior. In Figure 2g single A frequency, more bath realistic oscillators cause the condensed relatively oscillator frequencies, simple at phase ω j = behavior involves 9ε that are many in the not bath system. closeoscillators. This the spin behavior The system’s last of σ plot z natural coupled in Figure to a 2single4shows , cause bath the ing relatively oscillator from power the simple spectrum spin off of system behavior ω 0 of can into multiple be the in described the harmonic bath ≈ system. oscillators, ω o shows gives a responses over a range of frequencies and of greaterA more realistic magnitude. condensed Thisphase behavior involves needs as energy This many be behavior and bath passing studied coupled then oscillators. of from more σ zthe in coupled complex spin depth.The system, last to than aplot a singlesimple into with in the Figure sum all harmonic the of the 2 single oscillator powershown llator shows oscillator off frequencies Adding returning the of ωpower 0 and can single into except be spectrum then bath described spin returning for system. oscillatorsof ω = as multiple The 9ε into energy lower the frequencies, passing frequency spin system. from oscillation that the The are spin lower not system close frequency into the oscillation harmonic spin system’s is thenatural4. Thebath power oscillators, spectrumcoupled is morespectra to complex the– spin most than system, notably a simple by with the sum dip all in flow ofof spectra just below frequenciesbehavior spectra of of is just ν the σ = 0.1. z below flow coupled frequencies of to a singlethe and shownfrequency, single energy then frequencies oscillator flow returning back cause of energy andpower relatively into forth back the spectra from and spin simple forth spin – system. from most behavior to spin boson. notably The to in boson. lower the bysystem. the frequency dip This in the oscillationck bath and forth However, oscillatorB. except from for off spin for single ofLarger for ω to ω 0 boson. can oscillators beBaths = 9ε4. The power spectrum is more complex than a simple sum ofof ν = 0.1.described whereas ω jenergy is on ωpassing 0 cause from morethe complicated Coupling spin system many behavior. bath intooscillators the In harmonicFigure also changes 2 the behavior ofs.the single oscillator power spectra – most notably by the dip in spectra just below frequenciesver, ofoscillator Coupling ν for the = more single 0.1.oscillator complicated and many oscillatorsthen bath A ω jreturning more = behavior. where oscillators 9ε4realistic ≈ ω ω intoj In o shows also on Figure the condensed ωchanges spin0 gives cause 2 the system. amore phase the responses oscillator behavior complicatedThe involves over lower ofmany aσ frequency behavior. range z in bath the of the time frequencies oscillators. oscillationIn time Figure domain domain 2into is and The the a into of more last flow greater a physical plot of inresult. Figure Finally, 2 ismore atorenergy magnitude. at ωbackj = 9ε and This behavior needs to be studied more in depth.4 ≈ ω fortho showsfromgives spina aresponses to boson.Coupling physical result. many shows bath Finally, theoscillators power is thespectrum case alsowith changes over of many over multiple a range the aoscillators. range behavior of bath frequenciesbehavior is coupled and for needs of single greater to 25 oscillators bemagnitude. oscillators, studiedwhere more This spaced behavior in ω j depth. is evenly onneeds ω 0 cause between more 3ε complicated 5εof oscillators, frequencies Figure ofthe σcase z 3 in shows with coupled and the many of the time greater oscillators. case domain thewhere spin Figure into the system, 3 ashows with the case all where the thespin e. This system However,more physical result. shownFinally, frequencies behavior. In Figure 2the B. studied oscillator Larger more atin ω Baths depth. j = 9ε4 ≈ is ω theexcept case with for ω many = 9ε2and2. The of this4 oscillators. . The power Figure spin spectrum system 3 shows is iscoupled more the case complex to 25 where oscillators, than theaspaced simpleevenly sum between ofsystem spin system is obviously is coupled themore single tocomplex oscillator 25 oscillators, o shows than power just givesspaced spectra the a responses addition evenly – most between of over notably a new a range 3ε by oscillation, of frequencies as evident and of in greater the2and the 5ε dip2. The in the behavior spectra of of just this this system below is frequencies obviously morepower magnitude. spectra ofThis σ behavior needs to be studied more in depth.ger system Baths is obviously z ν (Figure = more 0.1. complex 3). It decays than just to a lower the addition energy state of a new over complex oscillation, time than and just the the asspin addition evident stays of in in a new the a oscillation, as evident inlow power energy spectra A more stateof until realistic σ z Coupling the (Figure LARGER condensed numerical many 3). ItBATHSsolution phase decays bath oscillators involves eventually a lower many also energy grows changes bath state unstable. oscillators. the the over power behavior Figure time spectra The and 3 greatly of the last σspin zplot (Figure resembles instays the Figure 3). in time It adecays domain 2 to a lower into aenergythe relowrealistic B. same shows energy Larger system the condensed state power more that Baths until spectrum is phase physical the presented numerical involves result. of multiple solution Makri’s many Finally, bath eventually solution, is the oscillators, oscillators. case 3 although grows with coupled The unstable. many alast more plot the careful spin comparison system, 2withstillall theeneeds power shown toAspectrum be frequencies made.more realisticof multiple except condensed for bath ω phase oscillators, = 9ε involves 4 . The many powerbath coupled spectrum oscillators.single except more oscillator realistic forstate oscillators. over Figure time and 3Figure greatly the spin 3resemblesstays shows in the a low case energy where state the until thethe same systemspin thatsystem is presented is coupled in Makri’s 25 oscillators, solution, 3 although spaced the spin is evenly more asystem, more complex between careful with than all 3ε thea simple 5εcomparison still sum ofquencies theA lastωplot= condensed power in 9εspectra – most notably by the dip in the spectra just below frequencies4.FigureThe2power phase showsspectrumthe involves poweris many spectrummorebath complexofnumerical solution eventually 2and2 grows . Theunstable. behavior Figure of this 3 greatly1oscillators. than a simple The last sumplot of in Figure 2needs2µto be made. system is obviously more complex than just theoscillator shows of multiple ν = the power 0.1. power bathspectraoscillators, spectrum – mostcoupled ofnotably multiple to the spinbybath thesystem,dip oscillators, within theall theresembles additionthe ofsame a new system oscillation, that is presented as evident in Makri’s in the solution,spectra coupledjust to below the spin frequencies system, with all the3j ω j. The other terms are all set initially to zero.V. Conclusion power spectra of σ z . shown shown Coupling frequencies many except bathfor oscillators ω = 9ε (Figure 3). It decays to a lower4. also The changes power spectrum the behavior isalthough energy a more state careful over comparison time andstill theneeds spin to stays be made. in alow energy state until the numerical solution eventuallyis more of σ complex grows z in the unstable.than time a simple domain Figuresum 3into greatlyofa resemblesV.lingthe Conclusionmany more single physical bathoscillatoroscillators result. power Finally, alsospectrachanges is the – most the casebehavior with notably many byof oscillators. theσ z dip inthethe Figure timespectradomain 3 shows justinto below the acase frequenciesBaths The model here has proven be non-trivial with interesting behavior. The approximationswhere the42 CMDITR Review of Undergraduate Research Vol. 2 No. 1 Summer 2005spin system is coupled to 25 oscillators, spaced evenly between 3ε 5εsicalofresult.ν = 0.1.of Single the same system Oscillators that is presented in Makri’s solution, 3 although a more careful comparison stillmade QHDFinally, seem is tothe givecase reasonably with many physical oscillators. results Figure while3suitably shows the 2 simplifying and case 2 where . The the thebehavior dynamics.of thisThe model here needs has to proven be made. to be non-trivial withem is system coupled Coupling is to obviously many 25 oscillators, bath more oscillators complex spaced also evenly thanchanges just between the the addition 3ε interesting behavior. The approximationsbehavior of 5εa new oscillation, as evident the2and2. ofThe σ z behavior in the time of this domain into amade in QHD seem give reasonably physical results while suitably simplifying the dynamics.more physical result. Finally, is the case with many oscillators. Figure 3 shows the case where the1e+06100000100001000Coupled Oscillators with ω < 9ε/4ω = 3ε/2ω = 7ε/4ω = 2ε/4ω = 17ε/8P (υ)P (υ)P (υ)P (υ)P (υ)1e+06100000100001000P (υ)Coupled Oscillator with ω = 9ε/4e 2: Power Spectra of Asymmetric Systems Coupled to Single Oscillators, using the sametions as Figure 1. The power spectra of the asymmetric systems show two major oscillations.h frequency oscillation, dependent on Ω and ε and the slower oscillation is dependent on ω.n oscillator where ω is the natural frequency of the spin system more complicated behaviors parameters. 3 The particular case studied with QHD is the asymmetric system, ε > Ω > 0atching a system, starting with σ x (0) = 1, relax. The resulting equations of the spin-bosonwere then numerically integrating using a 4th-order Runge-Kutta algorithm. The initialtions for the bath oscillators give minimal bath energy and uncertainty: p 2 j = ω jµ j2andr the gas phase (no bath oscillators), the system can be solved analytically, and σ z oscillates