5.74 RWF Lecture #9 and #109, 10 – 1From Quantum Beats to WavepacketsReading: Chapter 9.2.1-4, The Spectra and Dynamics of Diatomic Molecules, H. Lefebvre-Brion andR. Field, 2nd Ed., Academic Press, 2004.Last time:How to get a glimpse of mechanism: cause and effecte.g. The effect might be 〈N1〉t and the cause might be 〈h12〉t. ddtddttNh11212=−−()hωωrelation between cause and effector iddttthN1=Ω−Ω†transfer rate operatorToday: classes of pluck → classes of wavepacketQuantum Beats are usually simple, few-level coherences.Wavepackets are usually more complex, many-level coherences.Both are the result of a pluck where ∆t < h∆E .Kinds of pluck: * merely short* genuinely localizedA localized pluck prepares a zero-order non-eigenstate. The localized state character of this non-eigenstateis distributed (“fractionated”) over eigenstates that span an energy range ∆Elocalized.↑↓h/∆tpluck↑↓brightdark∆ElocalizedMIT Department of Chemistry5.74, Spring 2004: Introductory Quantum Mechanics IIInstructor: Prof. Robert Field5.74 RWF Lecture #9 and #10 9, 10 – 2If the pluck is sufficiently short, it prepares an a priori known initially localized excitation.“genuinely localized”If the pluck is not sufficiently short, it prepares an ill-specified coherent superposition“merely short”Classes of localization* polarization quantum beats (angularly localized)* population quantum beats (spatially localized or localized in state space, e.g., vibrationalwavepacket)Polarization QB - easiest kind to observe, even with a long excitation pulse. Why?((Hanle effect))32J′ M–1J′ M+1xyJM1J″MfluorescencepulsedexcitationI(t) = Trace (D U(t,0)Eρρρρ(0)E†U(t,0))xyyxDC Magnetic field in z directionExcite with x polarized light propagating in y directionDetect light propagating in z direction, IIIIxyxy−+.ZeemanSplittingz5.74 RWF Lecture #9 and #10 9, 10 – 3see polarization QB as B-field increases from 0.Larmor precession of magnetic moment about z direction. Transition dipole moves because it is attached tothe molecule frame and a magnetic moment that is fixed in the molecule frame precesses in thelaboratory frame.Has anyone looked at fluorescence from an atomic 1P — 1S transition excited by a linearly polarized laser?If the polarization axis of the exciting radiation is pointing toward you, you see no fluorescence. Thetransition moment is fixed in space.* can force the transition moment to rotate in external magnetic field* more complicated if there are other nonzero angular momenta* even more complicated if it is a molecule and not an atomPopulation QBE+ (Mostly T)E– (Mostly S1)S1TS0(dark)(bright)No polarization required. In fact, use geometry in which polarization quantum beat cannot be detected.What is that? Several schemes possible. (Alignment is a Tq2 second rank tensor quantity.)At t = 0 prepare S1 which is not an eigenstate. S1 is capable of fluorescing. T is not.As coherent superposition evolves, becomes predominantly T in character. Fluorescence rate decreasesbecause T cannot fluoresce.It is as if population flows back and forth between bright and dark state.21HST~5.74 RWF Lecture #9 and #10 9, 10 – 4Similarly for anharmonic couplingdarkbrightNow it is useful to discuss various kinds of localized excitations that are easily achievable.Pure rotation spectrum: microwave region (or TeraHertz)picture of I(ω):spectral width of microwave oscillator (10% bandwidth)JJ – 10∆E = 2BJrequires a permanent electric dipole momentJµradiation exerts a torque to transfer its 1 unit of angular momentummolecules with rJ pointing  to the radiation propagation direction (call it z) are preferentiallyexcited: M = ±Jexcitation is spatially anisotropic.Radiation polarized  µ is most effective in exciting moleculesddQDµ= 0ddQBµ≠ 05.74 RWF Lecture #9 and #10 9, 10 – 5–J+JMI∆M=±1∆J=±1–J+JMI∆M=±1∆J=±1JJ+1J–12B(J+1)2BJ↑↓↓↑TcB JROT=+121()and12cBJdipoles rotate relative to lab frameThere is a grand rephasing when all dipoles have returnedto their original orientation in the lab frame∆=TcBgrand rephasing12all J's undergo rephasing an integer number of timeseach ∆=tcB12B in cm–1Is it a physical picture ofdipoles rotating or only amathematical consequenceof rephasings in e–Et/h?5.74 RWF Lecture #9 and #10 9, 10 – 6Peter Felker: Rotational Coherence Spectroscopybased on characteristic grand (and sub-grand) rephasingssym. top EAKBJJ KABJK=+ +−[]>221()just a matter of a lot of integer-related rotating dipole antennas.Pump-probe schemes. Use of polarization in probe to capture grand rephasing.Vibration-Rotation SpectrumωVIB•–2BV 2B 4Bν32 1120P(J) R(J) B Branches J = I(ω)RJ B J J JJ B JBJ JPJ B J J JJ B J BJJ()=+()+()+()[]=+()=+ =()=−()+()[]=−()=−=12 1 222 1 012112212 3–() ,,–,,short period from A,longer period from B.zero gap5.74 RWF Lecture #9 and #10 9, 10 – 7PRat low resolutionzero gapνVibration-Rotation Spectrum: see picture of spectrum, I(ω)requires a change in electric dipole moment as molecule vibratesrrrµµµµµµQQQQQ()=()+∂∂−()=−∑ejNjjje136IR Active?linear CO2sym. stretch Q1bend (doubly degenerate) Q2antisym stretch Q3bent NO2sym. stretch Q1bend (non-degenerate) Q2antisym stretch Q3∆1 = ±1 strong fundamentalddQµ±2 weak overtonebothddQ22µ and mechanicalanharmonicity±3 very weakhigh overtones, especially for R-H stretches (n × 3000 cm–1)* parallel type, ∆K = 0*one for each K″* perpendicular type, ∆K = ±1*two for each K″5.74 RWF Lecture #9 and #10 9, 10 – 8brightn(R–H)darkStatistical limit Intramolecular Vibrational Redistribution(see supplement on Heller’s Fractionation Index, 10S-5)doorway (first tier)cubic or quartic anharmoniccoupling: ∆V = 3 or 4brightbathaccidentalresonance↑↓IVR, tiersBright, doorway, dark (bath)CH stretch: C–C–H bend 2:1 resonance (cubic anharmonicity, e.g. kQQ122122) — usually important. Ifthere are several near resonant coupling mechanisms, get multiple competing pathways.Doorway state sometimes can “dissolve” in dark bath.(very anharmonic, noother vibr. transitions ofcomparable intensity insame spectral region5.74 RWF Lecture #9 and #10 9, 10 – 9useful tools:aaaaRH RHt††() ( )≈Ψ Ψ02doorway doorwayThere will always be rotational recurrences in a pulse-excited vibration rotation spectrum. Do the rotationalrecurrences dephase when vibrational bright state dephases?