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Coherence and control in photo-molecular wave packet dynamics Han, Alex Chao 2015

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Coherence and Control in Photo-molecularWave Packet DynamicsbyAlex Chao HanB.Sc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2015c© Alex Chao Han 2015AbstractWave-mechanical phenomena such as resonance and interference, in bothlight and matter, are central to the principles of quantum coherent controlover molecular processes. Focusing on the dynamical aspects, this disser-tation is a compilation of studies on the interaction physics involving wavepackets in molecules, the driving light field, and the underlying coherenceand control. In each work, we will demonstrate interesting correlationsbetween the properties of a carefully designed excitation light field and de-sirable outcomes of the molecules quantum dynamics.We will analyze the dynamical effect of a Feshbach resonance in the adi-abatic Raman photoassociation for ultracold diatomic molecule formationfrom ultracold atoms. A narrow resonance is shown to be able to increasethe effective number of collisions, in an ultracold atomic gas, that are avail-able for photoassociation. This results in an optimal resonance width muchsmaller than the atomic collision energy bandwidth, due to the balancebetween the effective collision rate and single-collision transfer probability.Next, we demonstrate the linear molecular response to high-intensity, broad-band, shaped optical fields. We show that this originates from interferencesbased on intra-pulse Raman excitations, and thus response linearity is notunique to the first-order perturbative limit and can not be used to infer thestrength of the field. In the last study, we simulate the stochastic vibra-tional wave packet and dissociation-flux dynamics in a molecule excited bylight with temporal and spectral incoherent properties. Between this caseand that using a coherent pulse with the same spectral profile, we comparethe vibrational wave functions and the loss of electronic and vibrational co-herence, and demonstrate the qualitative difference between coherently andincoherently driven dynamics in molecules.iiPrefaceA large portion of materials in Chapter 3 was published in the article: A.C.Han, E. Shapiro, M. Shapiro, Pulsed adiabatic photoassociation via scatter-ing resonances, J. Phys. B: At. Mol. Opt. Phys. 44, 154018 (2011). ThisChapter is a continuation from many previous studies by Evgeny Shapiroand Moshe Shapiro, where a large part of the analytical framework (someof section 2 of the publication), and some computational tools, have alreadybeen developed. The author contributed in combining some existing butpreviously separate analytical models (section 2), and building extendedcomputational programs for these models (section 3). The author also car-ried out the numerical computations, whose results could not be triviallyseen from the analytical framework (section 3). Other parts of the pub-lication, not included in this thesis, have main contribution from EvgenyShapiro and Moshe Shapiro.Chapter 4 section 1 was published in the articles: A.C. Han, M. Shapiro,Linear response in the strong field domain, Phys. Rev. Lett. 108, 183002(2012), and section 2 in: A.C. Han, M. Shapiro, Linear response in thestrong-field domain: ultrafast wave packet interferometry in the continuum,J. Phys. B: At. Mol. Opt. Phys. 46, 085401 (2013). In the first publi-cation, Moshe Shapiro conceived the idea and analytical framework. Theauthor carried out the analytical derivation, chose the appropriate opticalshaping methods, developed the numerical computational tool and carriedout the numerical investigation, under the guidance of Moshe Shapiro. Thesecond publication is by invitation from Journal of Physics B, as a continu-ation of the first article. The two authors together developed the conceptualsetup, and the current author constructed the analytical time-domain modelas an extension to the frequency-domain one in first publication. The au-iiiPrefacethor developed the computational program, and conducted the numericalanalysis.Chapter 5 is based largely on the publication: A.C. Han, M. Shapiro, P.Brumer, Nature of quantum states created by one photon absorption: pulsedcoherent vs pulsed incoherent light, J. Phys. Chem. A, 117, 8199-8204(2013). Moshe Shapiro developed a theoretical framework based on conceptsconceived together with Paul Brumer. The author developed the analyticalderivation and the computational tools, and tested the hypothesis on variousmodel molecular potentials.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Theoretical background . . . . . . . . . . . . . . . . . . . . . . 72.1 Quantum structure and dynamics of molecules . . . . . . . . 72.1.1 Motion of a particle in central potential . . . . . . . . 72.1.2 Born-Oppenheimer states of molecules . . . . . . . . 102.2 Photoexcitation of molecules . . . . . . . . . . . . . . . . . . 132.2.1 Dipole transitions . . . . . . . . . . . . . . . . . . . . 132.2.2 Perturbation theory . . . . . . . . . . . . . . . . . . . 162.2.3 Principle of coherent control . . . . . . . . . . . . . . 192.2.4 Adiabatic processes . . . . . . . . . . . . . . . . . . . 232.3 Wave coherence . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Numerical computations . . . . . . . . . . . . . . . . . . . . 27vTable of Contents3 Adiabatic photoassociation of ultracold molecules via scat-tering resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Dynamics of quantum-state amplitudes . . . . . . . . . . . . 323.2.1 Equations for bound-continuum population transfer . 323.2.2 The effective modes expansion . . . . . . . . . . . . . 353.3 Numerical computation for Rb-Rb collisions . . . . . . . . . 403.3.1 Single-event transfer efficiency . . . . . . . . . . . . . 403.3.2 Ensemble transfer efficiency . . . . . . . . . . . . . . 443.3.3 Scaling behaviour with ensemble temperature . . . . 473.3.4 Thermalization and practical matters . . . . . . . . . 493.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Linear molecular response in the strong field domain . . . 524.1 Linear response via Raman transitions in a three-potentialmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Linear response in H+2 . . . . . . . . . . . . . . . . . . . . . . 634.2.1 Vibrational wave packet method . . . . . . . . . . . . 644.2.2 Response analysis . . . . . . . . . . . . . . . . . . . . 674.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 775 Quantum dynamics under incoherent photoexcitation . . 795.1 Electronic decoherence via random vibration . . . . . . . . . 795.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 795.1.2 Stochastic Schro¨dinger’s equation for random wavepacket . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1.3 Wave packet simulations and electronic decoherence . 875.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 935.2 Vibrational resonances excited by incoherent light . . . . . . 945.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 945.2.2 Excited-state dynamics of IBr . . . . . . . . . . . . . 955.2.3 Spectral incoherence . . . . . . . . . . . . . . . . . . . 975.2.4 Stationary wave packet and flux dynamics . . . . . . 100viTable of Contents5.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 1036 Final conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110viiList of Figures3.1 A schematic display of the Adiabatic Raman Photoassocia-tion (ARPA) process. Left panel: Atoms colliding in thenear-threshold energy range are excited by the “pump” laserto the vibrational state |2〉 on an excited electronic potential.The latter is coupled by the “dump” laser to the deeply boundtarget state |1〉. Right panel: The same setup but with theinclusion of a hyperfine-manifold bound level interacting withthe previous continuum, resulting in a Feshbach resonance. . 313.2 Left panel: Three bound states STIRAP. The populationwhich starts in the initial bound state |3〉 is transferred intostate |1〉 by following the evolution of the “dark” field-dressedstate. The process avoids population loss due to spontaneousemission from state |2〉 because the latter remains unpopu-lated. Middle panel: ARPA via a collisional resonance.The population gradually feeds the resonances as the contin-uum wave packet (shaded area) arrives (at time t0). Rightpanel: The resonance-dominated continuum of the middlepanel is replaced by a single (or several) effective mode(s)with decaying amplitude(s) B1(t) (Bs(t)). . . . . . . . . . . . 393.3 A single resonance is used for the computation, with magni-tude |µ2,E | = |µresΓres/ [2(E − Eres) + iΓres]| as a function ofenergy, shown for various choices of Γres and Eres = 100µK.The resonance height is µres = 300a.u., but can be made evenlarger to favour lower laser intensities. The Franck-Condonfactor involving eigenstate |E〉 follows the same shape. . . . . 41viiiList of Figures3.4 The target |1〉 and intermediate |2〉 state populations as afunction of time for two resonance widths. Top panel: thepump and dump field amplitudes. Middle panel: The targetstate population |b1(t)|2. The transfer efficiency is 90% forthe wide (100µK) resonance, but only 23% for the narrow(6µK) resonance. Bottom panel: The intermediate statepopulation |b2(t)|2. (Notice the large difference in the verticalscale relative to the middle panel.) . . . . . . . . . . . . . . . 433.5 Top panel: The target population, |b1(t → ∞)|2, for differ-ent resonance widths. Bottom panel: |b1(t → ∞)|2 as afunction of the centre of the resonance Eres, for E0 = 100µK;this shows transfer is optimal when the centre of the reso-nance coincides with the central energy of bE(0). . . . . . . . 443.6 The magnitude of the window functions fW (t) for differentresonance widths Γres and fixed height µres. Longer tails of|fW (t)| are observed for narrower resonances. . . . . . . . . . 453.7 Target state transfer efficiency |b1(t→∞)|2 for different res-onance widths as a function of the delay time δt ≡ t0 − tP ,where tP = 1.2µs is the pulses’ overlap peak time, and t0 isthe incoming wave packet peak time. . . . . . . . . . . . . . . 463.8 The time-averaged molecular production efficiency for a sin-gle pulse pair for an atom ensemble at 100µK, calculated byintegrating the delay plots of Figure 3.7. . . . . . . . . . . . . 463.9 Top panel: The single collision photoassociation efficiency|b1(t → ∞)|2 as a function of resonance width, for four dif-ferent values of the spontaneous decay rate. Bottom panel:The same plot at stronger laser intensities. The photoassoci-ation efficiency now becomes insensitive to the spontaneousdecay rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48ixList of Figures4.1 The model molecular system has two energy-degenerate ex-cited electronic potentials of continuum nature, coupled witha number of bound vibrational levels in the ground potentialby a broadband light pulse. . . . . . . . . . . . . . . . . . . . 534.2 Sample photo-products population dynamics for different timedelays and chirping parameters. The intensity value is µ2τI =0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 The photoexcitation yields or “response curves” (subplotsa,c) as functions of the field intensity, at particular shapingparameter choices, and the branching ratios as functions ofshaping parameters, at fixed intensities (subplots b,d). Theresponse curves are made more linear due to shaping. . . . . 594.4 The nonlinearity of the intensity response curves vs. µ2τI0. . 604.5 The intensity response curves for an initial superposition oftwo bound states ψi = c1 |E1〉+eiφ0c2 |E2〉. Upper two panelsare the intensity response and control for φ0 = 0 and variablec2; lower two panels are the response and control at fixedc22 = 0.4 and variable φ0. . . . . . . . . . . . . . . . . . . . . . 624.6 Schematic illustration for the photoexcitation of H+2 by an op-tical electric field with bandwidth covering three vibrationalstates of the ground 1sσg state. . . . . . . . . . . . . . . . . . 654.7 State populations as a function of laser peak intensity, andof delay times for various J-channels, and for the v = 3 andv = 7 states of the J = 0 channel. . . . . . . . . . . . . . . . . 704.8 State populations plots similar to Figure 4.7, but for a dif-ferent band of delay times approximately 20 fs later. Theresponse curves show almost identical behaviour. . . . . . . . 714.9 Similar state population plots as in Figure 4.7 and 4.8, butfor channels of higher J , with the same bands of delay times(same colour schemes for the plots). . . . . . . . . . . . . . . 724.10 Continued from Figure 4.9, for the later time band. . . . . . . 734.11 Branching ratio of populations in some pairs of vibrationaland rotational states are plotted, signifying coherent control. 74xList of Figures4.12 Population in various vibrational eigenstates v = 3 − 7 inthe J = 0 channel, and first three rotational channels, asfunctions of delay time tc. Superimposed are such scans withdifferent field intensities, with the same intensity range andcolour scheme as Figure 4.11. . . . . . . . . . . . . . . . . . . 754.13 Similar to Figure 4.12 but for higher rotational channels. . . . 764.14 The field-free probability-density for the initial vibrationalwave packet as a function of R and t. The beat frequency(≈ 5× 1013 Hz) is in agreement with the average vibrationalspacing (Ev=6 − Ev=4)/(2~). . . . . . . . . . . . . . . . . . . . 775.1 A realization of the incoherent light, and its magnified view,with central wavelength at ∼455nm (0.1 Hartree), and enve-lope function EL(t) ∼ [sin(pit/200 fs)]0.1 for 0 < t < 200 fs,and zero elsewhere, chosen to simulate CW behaviour with-out “sudden” numerical turning on and off. A random phasejump in the [−pi, pi] range, and a central frequency shift in the±0.0175 Hartrees range, is introduced every 7 fs on average. 855.2 The coherent pulse has transform-limited envelope functionEL(t) ∼ exp[−(t− 6 fs)2/2(1.6 fs)2]. A portion of the inco-herent pulse is also shown for comparison. The spectra of thepulses of coherent and a single realization of incoherent lightare also shown. The total energy flux,∫∞−∞ |E(t)|2dt, is keptthe same for the two types of pulses. . . . . . . . . . . . . . . 865.3 The spatio-temporal plots of the modulus-squared radial wavefunctions of the vibrational wave packets, ψ(1,2)x (R, t). Toprow is for case (A): purely repulsive excited state-1 and state-2 potentials, and bottom row is for case (B): bound Morsewells. The functional forms for both types of PES correspondto the first and third cases in Figure 5.5. . . . . . . . . . . . . 895.4 Same as Figure 5.3, but for photoexcitation with a singlerealization of the incoherent pulse. . . . . . . . . . . . . . . . 90xiList of Figures5.5 The left column shows three pairs of PES’s, and the v = 5vibrational wave function. The repulsive PES on the top hasfunctional form W (i)x (R) = W (i)0 exp[−(R−R(i)0 )/a(i)] +W(i)∞ ,and the Morse potentials in the bottom panel have W (i)x (R) =W (i)0[1− e−(R−R(i)0 )/a(i)]2 +W (i)∞ , both described by a param-eter vector vn = (W (n)0 , R(n)0 , a(n),W(n)∞ ). For the top panel,v1 = (0.1, 3.3, 1.0, 0), v2 = (0.1, 3.0, 1.2, 0.005). In the middlepanel, v1 is the same but v2 = (0.1, 2.5, 1.0, 0.01). In the bot-tom panel, v1 = (0.1, 5.0, 2.5, 0) and v2 = (0.1, 5.01, 2.53, 0.01).All numerical values are in atomic units. In the right columnis the absolute value of C(t), the normalized electronic coher-ence resulting from coherent and incoherent light excitationfor the three different PES of the left column. The resultsfor the incoherent light were averaged over a set of 10 realiza-tions. (Additional realizations did not significantly alter theresults.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.6 The potential energy surfaces for the ground, Y and B statesof IBr. Shown in the R = 23 − 25 Bohr range are the mag-nitudes of negative imaginary potentials, artificially added asnumerical absorbing boundaries, of functional form Vabsorb(R) =−i(0.075a.u.)(R− 23Bohr)2. . . . . . . . . . . . . . . . . . . . 965.7 Plotted in the top panel are the coherent (blue) and incoher-ent (orange) electric fields in time, and the modulation (t)(red, scaled). A pair of resonances, indicated by the peaks inthe φ0 → Y/B optical transition matrix element as a func-tion of energy, in the 17900-18000cm−1 range, is plotted inthe bottom panel (red). The power spectrum of the incoher-ent field (orange) is obtained from averaging over 100 fieldrealizations, whereas that of the coherent field (blue), ap-proximately 0.01614|s0(ω)|2, is simply scaled down from thepower spectrum of the incoherent source. . . . . . . . . . . . 99xiiList of Figures5.8 Sum of the magnitude-squared wave functions in the Y andB states, i.e. |ψY (t, R)|2 + |ψB(t, R)|2, are plotted in the topthree panels for the relevant fields. They are plotted in thesame spatial and temporal range, and the same colour scale.Vaguely shown is the extended portion for R > 8 Bohr. Butthey are clearly reflected in the plot for the flux in the lowerright panel. Middle panel is the flux as functions of time,created by component fields of the form [s0(ω)∆ω](t)e−iωt,spaced by ∆ω = 0.1cm−1. The sum of the flux across theExcitation frequency axis produces the total flux (red curve)in the panel below it. . . . . . . . . . . . . . . . . . . . . . . . 102xiiiAcknowledgementsI would like to, first and foremost, offer my most sincere gratitude to Pro-fessor Moshe Shapiro, my research supervisor guiding me for five valuableyears, an immensely acknowledgeable teacher, a highly productive and cre-ative colleague, and an invaluable mentor for life. From him, I’ve learnt notonly the scientific knowledge, but also a unique way of conducting research,communicating the science, insightful thinking and ideology, a sharp under-standing in most technical aspects, and a fearless attitude towards life. I’vehad the utmost privilege for being his student, and for knowing him as aperson and a friend. I finish this dissertation, covering all of our completedwork, with the deepest belief that his legacy will continue to influence allthe science he had helped developed, to encourage all the scientists he hadacquainted, and to inspire generations of young researchers to come.My most sincere gratitudes are offered to Professor Valery Milner, forhis endless effort to care for my academic and research work, all the waytowards my completion of the doctoral program. I’m grateful for the newand exciting research areas he has brought me into, his in-depth knowledgeand expertise in a range of fields and topics, invaluable guidance, a keen senseon both experimental and theoretical research, and everlasting enthusiasm,energy and passion for science.I would also like to thank Professor Kirk Madison, whom I’ve knownsince my very first course in quantum mechanics years ago. He has been anunprecedented teacher, with limitless passion for mentoring young genera-tion of scientists. Teaching with him had not only been my greatest pleasure,but had also immensely sharpened my own understanding and craft for thescience. Special thanks are also sincerely offered to Professor Roman Krems,who had provided me with invaluable advices in both academic and researchxivAcknowledgementsaspects, since the very start of my graduate studies, and to Professor Taka-masa Momose, who has greatly expanded my horizons of knowledge acrossphysics and chemistry, theories and experiments.I would like to offer my most sincere gratitudes also to Dr. EvgenyShapiro, Dr. Xuan Li, and Dr. Cian Menzel-Jones, all of who had taughtme, along with Moshe, essentially everything I needed to work efficientlywhen I started, and continued on to be among the most valuable colleagues Iever had. I thank all of them for their advices, in-depth knowledge, personalinsights and immense help to me personally, in the most detailed scientificand technical aspects, the communication and presentation of the science,and academic and research life at large. Special thanks are also sincerelyoffered to Dr. Asaf Eilam, whose scientific curiosity and creativity can nevercease to amaze us. I’d like to also thank all of my other dearest friends andcolleagues, from whom I have had received immense and endless advice,knowledge, inspiration and fun: Dr. Sida Zhou, Dr. Ping Xiang, Dr. JieCui, Dr. Marina Litinskaya, Dr. Ioannis Thanopoulos, Dr. Zhiying Li, Dr.Felipe Herrera, Dr. Yakun Chen, Devin Li, Alex Chow and Martin Bitter.xvTo my parents and my wifexviChapter 1Introduction1.1 MotivationThe wave nature of matter is one of the central concepts in modern physi-cal sciences. In particular, the quantum mechanical formulation of moleculeshas given rise to modern quantum chemistry, based on the Born-Oppenheimerstructure for the constituent electrons and nuclei. Quantization of physicalobservables, a direction implication of the quantum theory, not only re-sults in more accurate understanding of physical and chemical properties ofmolecules, at equilibrium and during the time-dependent evolution, but alsoopens up intriguing opportunities for experimentalists to actively influence,even to engineer, the outcome of physical and chemical processes, utilizingproperties unique to their wave nature, such as resonances and interferences.Among many approaches to tuning into the wave, or, as we will referto throughout this thesis, coherent, aspects of molecules, is the molecu-lar excitation using coherent electromagnetic wave, or simply light. Thisapproach is rather intuitive, considering mechanically that light, in particu-lar its time-varying electric field, exerts direct forces on most of constituentcharged particles in the molecule, such as positively charged protons and neg-atively charge electrons. More naturally existing light-molecule interactionphenomena range from vision in almost all animals, to the very life-creatingprocess of photosynthesis.The strongest interaction between electromagnetic wave and matter isthe action of the electric field on the molecule’s electronic and nucleardipoles. At appropriate optical frequencies, such interaction can bring themolecule very efficiently into other quantum states with different physicalproperties. If the molecule is also quite isolated from the environment, its11.1. Motivationstructure and dynamics, in the presence of the effects of optical radiation,can be effectively and efficiently analyzed and understood via basic waveformulation of quantum mechanics. The classical treatment of light andquantum mechanical formulation of matter is commonly referred to as semi-classical theory of light-molecule interaction.The principle of influencing molecular processes using electromagneticwaves is then sound, but the actual practice also requires the coherent ma-nipulation of the electric field itself. Thanks to modern inventions of laserlight sources, and technologies for the purposes of manipulating laser pulses,the laser excitation of molecules makes the coherent control of molecularprocesses feasible. Lasing, in its most basic form, signifies our understand-ing and control over the light generation. In a classical (as opposed toquantum-optical) sense, it is exactly because of the control over the phasedegree of freedom that gives us the generation of laser light: countless num-ber of smaller, time-varying electric fields at the same frequency oscillatesynchronously, to produce a very stable, large amplitude, sinusoidal totalfield. The extreme care given, in the first place, to produce such coherentlight would then be utilized for coherent control of molecular waves.The bandwidth of the excitation light, i.e. the range of frequencies or“colours” in its spectrum, suggests two categories for laser excitation ofmolecules. Considering the natural timescale for the motion of nuclei inmolecules is femtoseconds (10−15 second), laser pulses with duration a feworder of magnitude higher (e.g. nanoseconds) are considered and commonlyreferred to as “continuous wave”, or CW for short. In the spectral domain,the long duration translates to a narrow bandwidth. Thus, the energy uncer-tainty in the molecular excitation is narrow, in comparison with the energysplittings in the quantized energy structure of the molecule. In most casesthen, only single quantum eigenstate of a molecule is excited by CW light,while the phase of the light wave, imparted to the molecular wave functionas a total phase, is unobservable and irrelevant for the excitation outcome.However, the usage of CW light at more than one frequencies does enableus to influence molecular processes, via tuning the mutual phase betweenthe two components. The early theoretical and experimental demonstra-21.1. Motivationtions developed mainly by Moshe Shapiro and Paul Brumer, in the subfieldbetween physics and chemistry known as quantum coherent control, involvedtechniques based on such multi-colour interference effects.As technologies on producing and manipulating laser pulses with shorterand shorter durations mature, more researchers have turned to utilizedbroadband pulses to conduct coherent control studies. In most of today’soptics labs, it’s very convenient to produce short optical pulses with dura-tion of a few tens of femtoseconds, if not shorter. This is comparable tothe natural vibrational timescale of molecules, which, in the energy domain,corresponds to the pulse’s wide spectral band and the consequent ability toexcite a number of vibrational eigenstates of the molecule with, more impor-tantly, tuneable phase correlations. Such ultrashort, broadband, coherentpulses serve as an application of a large number of CW components, allexciting the molecule at the same time. Pulse shapers, either in the spec-tral or spatial domain, in particular, provides us with convenient spectralmanipulation of such collection of CW components. Such superpositions ofa large number of eigenstates, in both light and matter, corresponding toa large range of frequencies, are referred to as wave packets. So coherentcontrol, in particular for molecular processes involving wave packets, hasnever been such an easy task if not for ultrafast laser sources and methodsof pulse shaping.On top of these two categories of CW vs. broadband optical excitation ofmolecules, additional regimes of light-matter interaction are divided accord-ing to the intensity of the excitation light. Such classification is, however,not new or unique to optical excitations. Perturbation, in any physical form,is well understood in the quantum theory, and can be classified according toits effect on changing the original quantized structure of the matter it is ap-plied upon. We leave the detailed theoretical definition to the next section,since its physical interpretation can not stand alone without it. However, itis informative to outline that, with ever-increasing intensity of the opticalexcitation, the number of molecular eigenstates participating in any interfer-ence effect also increases. This can be due to the increased number of statesthe molecule can reach through various multiphoton processes, or, because31.2. Thesis outlineof the multiple times an eigenstate interfere with itself, while having reachedsome other intermediate states.A third dimension in the categorization of photo-molecular interactionbrings us back to the very quantification of coherence. If the phase cor-relation between the CW components of light, thus also between quantumeigenstates of molecules excited, is lost, what happens to the dynamics andcontrollability? It then becomes one of the goals of this thesis, to demon-strate some particular cases where we can very clearly observe the effects ofphotoexcitation of molecules under incoherent light.In summary, what really motivates modern research, in the context ofthis doctoral dissertation, on coherent control of the complex dynamics ofmolecules and many aspects of molecular processes, is based on (1) the wavenature of molecules, and its very intimate relation to optical coherence dur-ing its photoexcitation, and (2) how utilizing the interference, i.e. the phasedegree of freedom alone, can give us great influence over these molecularprocesses.1.2 Thesis outlineIn Chapter 2, we will first briefly outline the theoretical background asthe foundation upon which all studies of this thesis are rooted. We willdescribe the quantum mechanical structure and dynamics of molecules, andthe mathematical details of their interaction with electromagnetic waves.A few topics concerning different physical systems, but with commonunderlying physical mechanism, principles and strategies are then presentedin separate chapters. The first study of Chapter 3 aims to improve ultracoldmolecule production via photoassociating colliding ultracold atoms, utilizingscattering resonances. An optical excitation method rooted in the process ofSTIRAP (“STImulated Raman Adiabatic Passage”), has been developed byMoshe Shapiro prior to this study. But to also utilize the very versatile mag-netic manipulation of Feshbach scattering resonances in the atomic gases,we wish to update the scheme by analyzing the role resonances play bothin energy and time domains during the photoassociation. The result is both41.2. Thesis outlinecounter-intuitive and insightful: there exists an optimal resonance width,balancing the competition between large photoassociation rate with shortcollision durations (large resonance width) and lowered rate with stretchedcollision time (narrow resonance width).Two studies described in Chapter 4 share the same topic of linear molec-ular response in the strong-field regime of optical excitations. On the funda-mental level, they originate from some core ideas and importance of coherentcontrol, and the role it plays (actively or passively) in interesting physicalphenomena. They are based on the fact that, although a first-order per-turbative potential causes quantum state transition probabilities to dependlinearly on the perturbation strength, the converse statement is not true.Namely, a linear response curve does not exclusively come from a pertur-bative excitation. We demonstrate, in the optical case, that such linearitycan also come from a broadband, spectrally shaped, high-intensity field. Weintroduce various spectral shape modifications to the excitation pulse, bothamplitude-only and phase-only, so that the multiple excitation pathways in-terfere, and we show that what follows may drastically differ in the responselinearity.The inseparable link between light and matter is again of central focusin the fourth study described in Chapter 5. With the rise of “quantum bi-ology”, misconceptions and over-generalizations often occur in the boomingnew research area. Many heated debates ensued on whether experimentalfindings of long-lasting quantum coherence in light-harvesting molecule doexist in nature under sunlight. We approach this problem from the veryinitial stage of the physical process: the photo-absorption, and analyze elec-tronic coherence in simple molecules created by coherent vs. incoherentlight pulses. In the weak-field domain, an intuitive theory shows that themolecular vibration is directly and strongly influenced by the waveform ofthe incident light field that drives it, and hence acquires the same degree ofcoherence. Light as incoherent as sunlight would then create molecular vi-brations that destroy interference between Born-Oppenheimer states of themolecule, hence also its electronic coherence, with a much faster rate thanits coherent counterpart.51.2. Thesis outlineFinal conclusions are provided in the last chapter, both as a generalsummary overviewing the studies covered, as well as to outline and motivatefurther topics that can be studied in the general theme of coherence, control,and wave packet dynamics.6Chapter 2Theoretical background2.1 Quantum structure and dynamics ofmolecules2.1.1 Motion of a particle in central potentialCentral to the quantum mechanical formulation of matter is the time-dependentSchro¨dinger’s equationi~ ddt |Ψ(t)〉 = H |Ψ(t)〉 , (2.1)where, in Dirac’s notation, |Ψ(t)〉 is the time-dependent state of the physicalsystem that evolves in time, whose first time derivative is given as the ac-tion of the Hamiltonian operator on the state. When expanded as a series oftime-independent eigenstates of the Hamiltonian, multiplied with their re-spective time-dependent phase factor coefficients, the Schro¨dinger’s equationis effectively translated into a set of differential equations for the coefficients,using the orthonormality of the eigenstates,i~ ddt(∑ncn(t) |ψn〉)= H(∑ncn(t) |ψn〉)=∑ncn(t)En |ψn〉⇒ i~ ddtcn(t) = Encn(t) ⇒ cn(t) = cn(0)e−iEnt/~. (2.2)So, eigenstates to the total Hamiltonian of the system not only serve as thequantum state with well defined energies, but also become the natural firststeps to solve the dynamical problem.Particularly relevant to molecular systems is the problem for the mo-72.1. Quantum structure and dynamics of moleculestion of a particle of mass µ experiencing a potential energy V (r), in threedimensions. The Hamiltonian includes the kinetic and potential energies [1],H = pˆ22µ + V (r), (2.3)and has eigenstate wave functions satisfying[ pˆ22µ + V (r)]ΨE(r) = EΨE(r). (2.4)The spatial representation of the momentum operator, in spherical coordi-nate system, ispˆ22µ =12µ(pˆ2r +L2r2)pˆr =~i1r∂∂rr =~i( ∂∂r +1r), (2.5)where the squared total angular momentum operatorLˆ2 = −~2[ 1sin θ∂∂θ(sin θ ∂∂θ)+ 1sin2 θ∂2∂φ2](2.6)has eigenstate wave functions satisfyingLˆ2Y ml = ~2l(l + 1)Y ml , (2.7)which are commonly known as the spherical harmonics.A so-called central potential, which originates from conservative forcessuch as the Coulomb (electro-static) interaction force between charged par-ticles, has an isotropic functional form, i.e. independent from the angularcoordinates. We can then write V (r) = V (r), making the energy eigenvaluerelation[ 12µ(pˆ2r +L2r2)+ V (r)]ΨE(r, θ, φ) = EΨE(r, θ, φ). (2.8)82.1. Quantum structure and dynamics of moleculesIn this case, Lˆ2 commutes with pˆ2r and V (r) since neither has angulardependence, making it a compatible observable with the entire Hamiltonian.The consequence is that a state of the formΨE,l,m(r, θ, φ) = Y ml (θ, φ)φE(r)r , (2.9)must be a simultaneous eigenstate to both the Hamiltonian and L2. Then,the Lˆ2 operator will be replaced by its eigenvalue in equation (2.8), and thefactorization of 1/r will simplify the resulting equation[ pˆ2r2µ +~2l(l + 1)2µr2 + V (r)]φE(r)r= pˆ2r2µφE(r)r +1r[~2l(l + 1)r2 + V (r)]φE(r)= 1r[− ~22µ∂2∂r2 +~2l(l + 1)r2 + V (r)]φE(r)= EφE(r)r (2.10)to[− ~22µ∂2∂r2 + Ul(r)]φE,l(r) = EφE,l(r),Ul(r) =~2l(l + 1)2µr2 + V (r), (2.11)an effective 1D problem in the radial direction that describes the motion of amass µ, in angular momentum channel (l,m) with effective potential Ul(r),consisting of the original central potential plus a centrifugal term. Noticethat the quantum number m does not enter the effective Hamiltonian, hencenot determining the energy eigenvalues. So for brevity, the eigenstates arenot labelled by m, although in principle, do require its specification. Theeigenvalues to this problem will then be the energy eigenvalues to the origi-nal central potential problem, and the set of eigenfunctions φE,l(r)Y ml (θ, φ)solves any time-dependent motion of the particle in the central potential.92.1. Quantum structure and dynamics of molecules2.1.2 Born-Oppenheimer states of moleculesA molecule consists of a collection of electrons and nuclei, bounded in alocalized region of space, with certain geometric configurations. In order toanalyze the structure of a molecule, for example the geometric organizationof the nuclei or electronic wave functions and energies, we need to start bydefining its quantum mechanical Hamiltonian, and then solving it by findingthe energy eigenstates in which the molecule has a stable structure. Tothe strongest effect, electrostatic potential energies govern the interactionsbetween the constituent electrons and nuclei of the molecule. We thereforeinclude, in addition to the kinetic energies, only Coulomb interaction in themolecular HamiltonianHmol(r,R) = TˆN (R) + VNN (R) + Tˆe(r) + Vee(r) + VeN (r,R)=∑j−~22Mj∇2Rj +e24pi0∑j 6=kZiZj|Rj −Rk|+ −~22m∑j∇2rj+ e24pi0∑j 6=k1|rj − rk|+ e24pi0∑j,k−Zj|rj −Rk|, (2.12)which has energy contributions from kinetic energies of nuclei TˆN (R) andelectrons Tˆe(r), electrostatic energy between nuclei VNN (R), between elec-trons Vee(r), and between nuclei and electrons VeN (r,R). The set of allnuclear coordinates is denoted by shorthand notation R = {Rj}, with jlabelling each nucleus, and electronic coordinates by r = {rk}, with k la-belling each electron. The effective electronic Hamiltonian then includes allterms involving the electrons,He(r,R) = Tˆe(r) + VNN (R) + Vee(r) + VeN (r,R), (2.13)so thatHmol(r,R) = He(r,R) + TˆN (R). (2.14)The Born-Oppenheimer formulation then proceeds to first solving theelectronic problem, by looking for the electronic eigenstates to the electronic102.1. Quantum structure and dynamics of moleculesHamiltonian,He(r,R)Φn(r,R) = Wn(R)Φn(r,R) (2.15)where the electronic energy eigenvalue is expected to be dependent on thenuclear coordinates. The electronic energy eigenvalues Wn(R) are there-fore a function of R as well, and are also known as the Potential EnergySurfaces (PES), for the electronic state labelled by n. Depending on thenuclear degree of freedom of the molecule, namely the number of coordinatevariables in set R, these PES’s in general are multi-dimensional. Therefore,such “electronic structure” problem becomes an essential step towards ana-lyzing any structural and dynamic properties of the molecule. In practice,however, the main research effort on tackling the electronic problem liesin the branch of quantum chemistry, where systematic and sophisticatedanalytical and computational methods have been developed. For practicalconsiderations of this thesis, much of the electronic structural informationis therefore obtained using quantum chemistry softwares, or assumed fromprevious studies.Assuming such electronic problem is solved, i.e. the electronic wavefunctions and the potential energy surfaces are known, then the well-knownBorn-Oppenheimer approximation asserts that the total energy eigenstatewavefunction Ψ(r,R) to the total molecular Hamiltonian is separated into anuclear and an electronic part,Ψn,E(R, r) = χE(R)Φn(r,R). (2.16)Substituting it into the eigenvalue relation for the total Hamiltonian, wehave an equation for the nuclear partHmol(r,R)Ψn,E(R, r) = [TˆN (R) +He(r,R)]χE(R)Φn(r,R)= [TˆN (R) +Wn(R)]χE(R)Φn(r,R)= EχE(R)Φn(r,R). (2.17)The adiabatic approximation within the Born-Oppenheimer formulation112.1. Quantum structure and dynamics of moleculesnow assumes the electronic states are such that∫Φ∗n(r,R)TˆNΦn(r,R)dr = 0,simplifying the above equation to a pure nuclear one,[TˆN (R) +Wn(R)]χE(R) = EχE(R). (2.18)This equation becomes especially clear for diatomic molecules, where,upon changing to the centre of mass frame, the nuclear kinetic operator isfor the reduced mass µ, and the PES is a function of only the bond lengthmagnitude R = |R|,[− ~22µ∇2 +Wn(R)]χE(R) = EχE(R). (2.19)This amounts to the exact setup of the central potential problem, wherethe eigenstate solutions are known to be χn,l,ν(R) = φn,l,ν(R)Y ml (θ, φ), withφn,l,ν(R) being the vibrational eigenstate to the 1D problem[− ~22m∂2∂r2 + Un,l(r)]φn,l,ν(r) = Eνφn,l,ν(r), (2.20)with effective potentialUn,l(R) = Wn(R) +~2l(l + 1)2µR2 .So, a full energy eigenstate of a molecule can then be uniquely referredto as the ν’th vibrational level, belonging to the electronic state n androtational channel l (sometimes symbol J is used in the place of l), in theabsence of other mechanisms that may add further quantum numbers. Wecan then also write the wave function of the eigenstate asΨn,l,ν(r,R) =φn,l,ν(R)R Yml (θ, φ)Φn(r,R). (2.21)In some situations we analyze in this thesis, only a small number of122.2. Photoexcitation of moleculesBorn-Oppenheimer energy eigenstates of diatomic molecules are relevant tothe physical process. In this case, the time-dependent quantity will be infact the complex amplitude associated with each eigenstate, subject to thelight-molecular interaction dynamics that we will outline in the next section.However, some cases can involve a large number of eigenstates, or continuumeigenstates which need to be included for a continuous range of energies.Then the direct analysis of the dynamics of complex amplitudes become lesstransparent, comparing with a rather more concise and convenient approachto simply combine vibrational states within the same rotational-electronicchannel into a vibrational wave packetΨn,l(r,R) =ψn,l(t, R)R Yml (θ, φ)Φn(r,R). (2.22)The time dynamics would be accounted for within the motion of the wavepacket, and the initialization of molecular population in various eigenstates,would translate to the specification of spatial wave function for the initialvibrational wave packet.The advantage of analyzing vibrational wave packets will become clearin the study in chapter 4 and 5, where spatial-temporal insights can begained from looking at the wave packet dynamics. However, the wave packetmethods does pose computational constraints, since for long times, caremust be given to its spatial divergence, a problem absent when using energyeigenstate amplitudes.2.2 Photoexcitation of molecules2.2.1 Dipole transitionsThe Born-Oppenheimer structure of molecules tells us that the complexstructure of molecules will be reflected in the additional quantum numberslabelling its full energy eigenstates. They refer to many degrees of freedom ofmolecular motions, such as vibration and rotation of nuclei, and motions ofthe electrons. Their wave functions can be, in principle, obtained via stagesof solving the electronic and nuclear Hamiltonian problems. Once known,132.2. Photoexcitation of moleculesthese eigenstate wave function, in turn, would be very useful in analyzinginteraction scenarios between the molecule and other mechanisms that caninduce transitions between these Born-Oppenheimer states.Central to all studies in this thesis is the photoexcitation of molecules.Due to the large difference in length scales, the electric field E(t) of theexcitation light (with wavelength at hundreds of nanometers) is considered tobe spatially uniform across the molecule (sizes of angstroms to nanometers).To the strongest effects [2], the electric field interacts with the molecule’stotal electric dipole µ (a spatial vector). This calls for an additional term inthe Hamiltonian of the system, describing light-molecule interaction energyH = Hmol − µ ·E(t), (2.23)where the total dipole of the molecule consists of dipoles from each of theconstituent electrons and nucleiµ =∑j(−erj) +∑jeZjRj = µe(r) + µN (R), (2.24)where we use shorthand notation r and R to refer to the sets of electronicand nuclear position vectors.This additional term’s time-dependence comes only from the time-varyingelectric field of light, which we require on physical ground to satisfy E(t→±∞) = 0. The physical picture is then, for times long before and long afterthe photoexcitation, the asymptotic total Hamiltonians are simply Hmol, themolecular Hamiltonian. During the light pulse, the molecule makes transi-tions between the energy eigenstates, because of the previously non-existingcoupling between the energy eigenstates now provided by the field-dipoleinteraction term.Therefore, it becomes essential to obtain the matrix elements of theinteraction Hamiltonian, which is proportional to those for the dipole op-erator µ, in the basis of the molecular energy eigenstates |Ψn,l,ν〉 to Hmol.Without writing out the full coordinate dependence, we only separate themolecular eigenstate into the electronic and nuclear parts as |Ψn,l,ν〉 =142.2. Photoexcitation of molecules|χn,l,ν(R)〉 |Φn(r,R)〉, and calculate the matrix elements of the interactionas〈Ψn′,l′,ν′ |µ |Ψn,l,ν〉=∫ ∫ [χn′,l′,ν′(R)Φn′(r,R)]∗µ(r,R)[χn,l,ν(R)Φn(r,R)]dRdr=∫χ∗n′,l′,ν′(R){∫Φ∗n′(r,R)µe(r)Φn(r,R)dr}χn,l,ν(R)dR+∫χ∗n′,l′,ν′(R)µN (R)χn,l,ν(R)[ ∫Φn′(r,R)Φn(r,R)dr]dR.(2.25)This very explicit expression for the matrix elements, especially off-diagonal ones with (n′, l′, ν ′) 6= (n, l, ν), classifies dipole-induced transitionsinto two types:(i) Vibrational transitions between electronic states, when n 6= n′. Theinner electronic integral between orthogonal electronic wave functions thenmakes the second term zero. So matrix element is determined solely bythe first term, involving the integration of the two vibrational wave func-tions, and the electronic transition dipole moment, also known as transitiondipole function. In many molecules, such transition dipole moment func-tion is varying very slowly with nuclear coordinates R, comparing with thevibrational functions themselves. Such situation gives rise to the Franck-Condon approximation, where the transition dipole function is takenout of the integration over R, and the matrix element is proportional to theproduct Dn′,nF(ν′,l′),(ν,l), where Dn′,n is the approximated transition dipolefunction, and F(ν′,l′),(ν,l) is the Franck-Condon factor between the twovibrational levels. This type of transitions is the focus of this thesis.(ii) Pure vibrational transitions within a single electronic state, whenn = n′. In this case, the inner integral in the second term between the elec-tronic wave functions is nonzero. Thus the entire second term contributes,and represent, for the off-diagonal elements, the transitions due to the nu-clear dipole. The electronic integral in the first term, on the other hand, nowinvolves the same electronic wave function for the electronic state labelled152.2. Photoexcitation of moleculesby n. For homonuclear diatomic molecules, the symmetries of the electronicwave function combined with that of the electronic dipoles make such in-tegral zero, whereas it is not zero in principle for heteronuclear diatomicmolecules.A great part of the rich content of photo-molecular physics and chem-istry is based on our understanding of and direct access to the details of suchdipole-induced transitions in molecules. While conventional spectroscopyaims to investigate and discover molecular properties based on the observa-tions from light emissions following these transitions, our ability to influencethe light field, thus also the matrix elements of the light-molecule dipole in-teraction Hamiltonian, can also result in many selective manipulation ofmolecular processes. In the following sections, we will present some theoret-ical demonstrations of such control ability.2.2.2 Perturbation theoryPerturbation theory is a mathematical framework that can be especially ben-eficial for the analytical calculations for light-induced transitions in molecules.The general setup for the time-dependent formulation starts, again, by iden-tifying a molecular part and interaction part in the total Hamiltonian, as inequation (2.23),H(t) = Hmol +HI(t),where Hmol has Born-Oppenheimer eigenstates |Ej〉 with shorthand indexj including all relevant quantum numbers.Assuming a general molecular state as an expansion|Ψ(t)〉 =∑jbj(t)e−iEjt/~ |Ej〉 , (2.26)the Schro¨dinger’s equation for the total Hamiltoniani~ ddt |Ψ(t)〉 = H(t) |Ψ(t)〉 , (2.27)can be rewritten, using the molecular and interaction parts of the Hamilto-162.2. Photoexcitation of moleculesnian, in the so-called interaction picture asi~ ddtb(t) = HI(t)b(t), (2.28)with eigenstate amplitude vector[b(t)]j = bj(t) (2.29)and interaction Hamiltonian matrix[HI(t)]ij = 〈Ei|HI(t) |Ej〉 ei(Ei−Ej)t/~. (2.30)Then, the perturbation expansion proceeds by presenting a recursivesolution for the state amplitude vector in the formb(t) = b(−∞) + −i~∫ t−∞HI(t1)b(t1)dt1, (2.31)where we see the value of b(t) at time t depends on not only its initialcondition b(−∞) but also its entire history from the beginning until time tinside the integral. Then, repeatedly applying this formula to the “history”state amplitudes inside the integral, we can generate the infinite seriesb(t) = b(−∞) + −i~∫ t−∞HI(t1)[b(−∞) + −i~∫ t1−∞HI(t2)b(t2)dt2]dt1= b(−∞) + −i~∫ t−∞HI(t1)b(−∞)dt1+(−i~)2 ∫ t−∞HI(t1)∫ t1−∞HI(t2)b(t2)dt2dt1= b(−∞) + −i~∫ t−∞HI(t1)b(−∞)dt1+(−i~)2 ∫ t−∞HI(t1)∫ t1−∞HI(t2)b(−∞)dt2dt1+(−i~)3 ∫ t−∞HI(t1)∫ t1−∞HI(t2)∫ t2−∞HI(t3)b(t3)dt3dt2dt1= ... (2.32)172.2. Photoexcitation of moleculeswhich can compactly written asb(t) =[1 +∞∑n=1U(n)(t)]b(−∞),U(n)(t) =(−i~)n ∫ t−∞∫ t1−∞. . .∫ tn−1−∞HI(t1)HI(t2) . . .HI(tn)dtn...dt1.(2.33)In most cases, such Dyson series [3] converges, with higher n terms con-tributing negligibly to the values of b(t). Accordingly, we refer to U(n)(t) asbeing responsible for the “n’th order” perturbation or transition.With this formulation, we can analyze a very particular type of thephotoexcitation of a “ladder” N -state system, where HI has nonzero elementonly for neighbouring states, and initial population is completely in |E1〉,i.e. b(−∞) = (1 0 . . . 0)T ≡ eˆ1 in the basis of {|E1〉 , |E2〉 ... |EN 〉}. Inthis case, HI can in fact be written in the form of “ladder operators” due toits matrix element arrangement, HI(t) = A†(t) + A(t), where A(t) has onlynonzero element at its lower sub-diagonal. We can then calculate arbitrarybm>1(t) asbm(t) = eˆm[1 +∞∑n=1U(n)(t)]eˆ1 = eˆm∞∑n=1U(n)(t)eˆ1 = eˆm∞∑n=mU(n)(t)eˆ1(2.34)where unit vector eˆm has 1 at element m and 0 elsewhere. The change inthe summation is because only perturbation terms higher than m includeproduct of m or more “raising” ladder operators A†(t), in the expansion inequation (2.33), that can connect eˆ1 (|E1〉) to eˆm (|Em〉).As a result, for such “ladder climbing” excitation scenarios, we can quan-tify the strength of the interaction Hamiltonian, e.g. the intensity of the ex-citation field incident to a molecule, according to its influence on the magni-tude of population |bm(t)|2 for molecular eigenstate |Em〉. The comparisonof population in each eigenstate is in fact very efficient and unambiguous,since the system is closed (or possibly open only to leaking of population)and total population is normalized to unity. Any further interference effect182.2. Photoexcitation of moleculesinvolving the amplitude or populations would not change such comparison.This is going to be quite relevant to our study in Chapter 4, in terms ofdetermining the strength of the field. “Weak field” is used interchangeablywith a “first-order perturbative field”, where excited populations of molecu-lar eigenstates in the “ladder” higher than the second one are much smallerthan unity and thus negligible. A “strong field” refers exclusive to a fieldstrong enough that perturbations with order higher than 2 cause significantpopulation transfer. Conversely, if we found significant population excitedto the 3rd eigenstate in the “ladder” or higher, the field is then definitivelyqualified as a strong field.2.2.3 Principle of coherent controlThe area of study on coherent control, in principle, is to utilize the opticalphases of the excitation field, to achieve selectivity in the molecule’s tran-sition among the energy eigenstates. In order to gain some quantitativeunderstanding about how the transition dynamics can be influenced by var-ious properties of the excitation field, let’s next analyze an explicit situation.Consider only two Born-Oppenheimer energy eigenstates, for simplicity la-belled as |0〉 and |1〉, of a diatomic molecule interacting with a pulse of laserlight E(t). Although in practice, such field is expected to be convergent intime, i.e.E(t→ ±∞) = 0, it is however theoretically instructive to first ana-lyze the more elemental molecular excitation by monochromatic field of theform E(t) = Eˆ0 cos(ωLt+ φ). Here Eˆ is its polarization direction, 0 is theamplitude, and ωL is the angular frequency of the light assumed to be veryclose to the energy separation between the two states (ωL ∼= |E1 − E0|/~).The Schro¨dinger’s equation, in the basis of |0〉 and |1〉, can then bewritten asi~ ddt(b0(t)e−iE0t/~b1(t)e−iE1t/~)=[(E0 00 E1)−(0 µ12µ21 0)0 cos(ωLt+ φ)](b0(t)e−iE0t/~b1(t)e−iE1t/~), (2.35)192.2. Photoexcitation of moleculeswhere E0,1 are the energy eigenvalues for the two states, and µij = 〈i| µˆ |j〉 =〈i|µ · E |j〉 are the matrix elements of operator representing the moleculardipole in the field’s polarization direction, in the same basis. The diagonalelements for the interaction are commonly assumed to be zero, which canbe for example due to wave function parity in molecules. Notice that, here,the amplitudes for the two states have time-dependence in addition to thephase factor of time as found in equation (2.2), due to the interaction term.Carrying out the time differentiation, the above equation simplifies toddt(b0(t)b1(t))= i~0 cos(ωLt+ φ)(0 µ01e−iω10tµ10eiω10t 0)(b0(t)b1(t)), (2.36)where we define the shorthand notation ω10 = (E1 − E0)/~.It is common, at this stage, to make the so-called rotating-wave ap-proximation for the excitation field, to further simply the above equation.We retain only the term with smallest oscillation frequency, i.e. the eiωtterm with smallest |ω|, on the right hand side. The cosine function can beexpressed as cos θ = (eiθ + e−iθ)/2. The reason for such approximation is aquantitative one: Considering the general solution given as the perturbationexpansion from equation (2.33), each U(n)(t) would have matrix element inthe form∑j∫ t0e−iωjt′dt′ =∑je−iωjt′−iωj∣∣∣∣t0=∑j1− e−iωjtiωj, (2.37)where we can see, the largest contribution comes from the term in the sum-mation with greatest 1/ωj , i.e. smallest ωj .In equation (2.36), the smallest oscillation frequency is |ωL − ω10|, sinceωL is assumed to be near-resonance with the state energy separation. Fora numerical example, the central laser frequency used to optically couple aground vibrational state to an excited one, for our study on IBr moleculein chapter 5, is around 538 THz. But the value for “|ωL − ω10|”, i.e. possi-ble laser frequency’s detuning from the energy separation between the twostates, is no more than 3 THz. This results in (1/3)/(1/538) ∼= 180 times202.2. Photoexcitation of moleculesbigger contribution from the small-frequency component.Therefore, with the rotating-wave approximation, and denoting ∆ω =ωL − ω10, equation (2.36) effectively becomesddt(b0(t)b1(t))= i02~(0 µ01ei∆ωt+iφµ10e−i∆ωt−iφ 0)(b0(t)b1(t))= i02~(µ01ei∆ωt+iφb1(t)µ10e−i∆ωt−iφb0(t)). (2.38)If the molecule completely resides in state |0〉 at t = 0, and the electricfield is first-order perturbative, we can assume for all subsequent times thatdb0(t)/dt ∼= 0, hence b0(t) ∼= 1 ([1] and section 2.2.2). This way, aboveequation gives us, in the case of nonzero detuning ∆ω 6= 0,ddtb1(t) =i02~ µ10e−i∆ωt−iφ⇒ b1(t) =i0µ102~ e−iφ∫ t0e−i∆ωt′dt′ = 0µ102~∆ωe−iφ(1− e−i∆ωt).(2.39)The oscillation of the excited state population is then|b1(t)|2 = 20|µ10|2/~2∆ω2. (2.40)For ∆ω = 0, we haveddtb1(t) =i0µ102~ e−iφ ⇒ b1(t) =( i0µ102~ e−iφ)t (2.41)where the rate of linear increase in |b1(t)|2 is 20|µ10|2/4~2. The linear depen-dence in time is a consequence of the assumption that b0(t) ∼= 1 for all times,which, of course, will be unphysical at arbitrarily long time. Therefore, thissolution is for short time only,In both cases, the characteristic rates of change in the excited state pop-ulation are proportional to the intensity of the excitation field 20 and thestrength of the transition dipole moment |µ10|2, both very reasonable since212.2. Photoexcitation of moleculesa stronger excitation, or a stronger coupling would be expected to increasetransition rate. However, in either case, the transition rates are insensitiveto the initial phase of the field φ, a coherence property of the excitationfield. Therefore, varying the phase of the electric field of the perturbative,monochromatic excitation light would not affect the transition dynamicsbetween two energy eigenstates. This forms the basis of the principle thatthere can be no phase-only coherent control, when a single eigenstate is ex-cited by a first-order perturbative field. If the single eigenstate is excited bya broadband pulse, the same derivation as above follows for each monochro-matic component of the pulse, coupling state 0 to a different excited state,thus reconfirming such no-control principle in the first-order perturbativedomain. We will discuss such principle again as the motivation behind thestudy in Chapter 4.To get around the no-control scenario with first-order perturbative field,one can either utilize another field with second order perturbative strength,amounting to the classic 1- vs. 2-photon coherent control technique, or,excite from an additional source state using first-order perturbative field,which is known as bichromatic control scheme [4]. Both cases rely on theestablishment of multiple “excitation pathways” to the same target excitedstate, and can offer us control over the transition dynamics via only thephases of the fields. For the bichromatic control, assume the excitationof |1〉 involves an additional source eigenstate |0′〉, which for example canbe a different vibrational eigenstate with all quantum numbers identical to|0〉 except vibrational. Accordingly, another electric field E′(t) appropriatefor their energy separation is used. The direct application of the aboveperturbation analysis would give us, for the nonzero detuning case,b1(t) =0µ102~∆ωe−iφ(1− e−i∆ωt) + ′0µ10′2~∆ω′ e−iφ′(1− e−i∆ω′t). (2.42)222.2. Photoexcitation of moleculesThe oscillation in the excited population is then (assuming µij are real)|b1(t)|2 =∣∣∣∣0µ102~∆ωe−iφ(1− e−i∆ωt) + ′0µ10′2~∆ω′ e−iφ′(1− e−i∆ω′t)∣∣∣∣2=∣∣∣∣0µ102~∆ω∣∣∣∣2sin ∆ωt2 +∣∣∣∣′0µ10′2~∆ω′∣∣∣∣2sin ∆ω′t2 + 20µ102~∆ω′0µ10′2~∆ω′ ·sin ∆ωt2 sin∆ω′t2 cos[(∆ω −∆ω′)t2 + (φ− φ′)], (2.43)where, if the field frequencies are chosen such that ∆ω−∆ω′ = 0, the relativephase angles φ−φ′ between the to fields can, now, alter the amplitude of thepopulation oscillation. The at-resonance case is even more clearly dependenton the relative phases:b1(t) =( i0µ102~ e−iφ + i′0µ10′2~ e−iφ′)t⇒ |b1(t)|2 =∣∣∣∣i0µ102~ e−iφ + i′0µ10′2~ e−iφ′∣∣∣∣2t2=[∣∣∣∣0µ102~∆ω∣∣∣∣2+∣∣∣∣′0µ10′2~∆ω′∣∣∣∣2+ 2 0µ102~∆ω′0µ10′2~∆ω′ cos(φ− φ′)]t2.(2.44)This demonstrates the principle of how altering the relative phases be-tween two monochromatic field components, connecting two different Born-Oppenheimer states to the same target state, can give us control over thetransition dynamics in a molecule. Such interference effects between pho-toexcitation pathways will be discussed in many places in this thesis.2.2.4 Adiabatic processesWhen the excitation field is pulsed, i.e. convergent for t → ±∞, and themodulation varies very slowly with time, the adiabatic theorem of quantummechanics [1] tells us the system will remain in its time-dependent energyeigenstate throughout the process. Transition processes in molecules satis-fying such condition are known as adiabatic processes.232.2. Photoexcitation of moleculesOne particularly successful technique of adiabatic manipulation of quan-tum state transitions in molecules is known as the stimulated Raman adia-batic passage (STIRAP) [5]. With STIRAP, the molecule under the excita-tion of two pulsed laser fields can transition between two energy eigenstates,say |0〉 and |2〉, with no direct optical coupling, i.e. µ02 = 0, by utilizinga third state with coupling to both. The appropriate design of the pair ofexcitation pulses can result in 100% transfer, namely, without the excitationof the intermediate state at all times.We briefly outline the mathematical reasons as below. The field-dipoleinteraction Hamiltonian matrix in the basis of {|0〉 , |1〉 , |2〉}, under rotating-wave approximation, isHI(t) =0 V01(t)eiω01t 0V ∗01(t)e−iω01t 0 V12(t)eiω12t0 V ∗12(t)e−iω12t 0 (2.45)where the two electric fields are assumed to be at-resonance with the energyseparations, ω01 = (E1 − E0)/~ and ω12 = (E2 − E1)/~. The dynamicalequation for such three-state problem can be derived in a very similar wayto the derivation from equation (2.35) to (2.36), but now we would haveddtb0(t)b1(t)b2(t) = i~0 V01(t) 0V ∗01(t) 0 V12(t)0 V ∗12(t) 0b0(t)b1(t)b2(t) . (2.46)Given that the field modulation Vij(t) varies slowly enough for the processto be adiabatic, the system remains in one of the time-dependent eigenstates|E±(t)〉 =1√2± 1√(V12(t)/V01(t))2+11± 1√(V01(t)/V12(t))2+1 , |E0(t)〉 =1√(V01(t)/V12(t))2+101√(V12(t)/V01(t))2+1 ,(2.47)242.3. Wave coherenceto the total time-dependent Hamiltonian, with time-dependent eigenvaluesE±(t) = ±√V 201(t) + V 212(t), E0 = 0.Key to the optimal |0〉 to |2〉 transition is the so-called “counter-intuitive”arrangement of excitations, where V12(t) proceeds V01(t), ensuring V12(t)V01(t) as t→ −∞, and V12(t) V01(t) as t→∞. This way,|E±(t→ −∞)〉 ∼=1√2011 , |E0(t→ −∞)〉 ∼=100 , (2.48)and after the excitation,|E±(t→∞)〉 ∼=1√2110 , |E0(t→∞)〉 ∼=001 . (2.49)The net effect is that, if the initial state of the molecule is entirely |0〉, themolecule occupies the |E0〉 state in the time-dependent picture. So whenthe interaction is finished, the molecule, remaining in |E0〉, is now entirelyin |2〉. An “adiabatic passage” from |0〉 to |2〉 is thus formed using the time-dependent energy eigenstate |E0〉. In our first study, in Chapter 3, we willencounter a similar three-state transfer problem in the context of ultracoldmolecule production. We will extend such STIRAP technique, to includecontinuum eigenstates with resonances.2.3 Wave coherenceFor either matter waves, arisen from the quantum mechanical formulation,or the electromagnetic waves which are solutions to the Maxwell wave equa-tion, eigenstate decomposition technique for solving the dynamics of thewave packet, similar to equation (2.2), can be applied. For so-called lin-ear problems, the complex amplitudes for the eigenstate evolves trivially in252.3. Wave coherencetime, according to phase factor e−iωt where ω is the characteristic frequencyof the eigenstate describing energy eigenvalues for matter, and angular fre-quency for light. For problems involving eigenstate transitions, dynamicscan also be solved by analyzing the coupling between amplitudes of eigen-states, i.e. calculating the off-diagonal matrix elements of the term in theHamiltonian responsible for transition among eigenstates. In this case, boththe magnitude and the phase of the complex amplitudes evolve in time.In both light and matter, essential to controlling the time evolution ofwave packets of eigenstates is the interferences between the complex ampli-tudes. We have conducted analysis, so far, under the assumption that theamplitudes are deterministic variables. This gives us full information aboutthe relative phases between eigenstate amplitudes, at any given time. How-ever, in order to analyze situations of incoherence, another important con-cept for later studies in this thesis, in both light and matter, we consider eachamplitude’s evolution as a stochastic one, meaning that the complex ampli-tude is modelled by random variables instead. Effectively, the quantum statevector, written in certain basis of dimension N as |Ψ〉 =(c1 c2 . . . cN)Talso becomes a random vector.The implication of such incoherence is often loss of mutual coherencebetween eigenstates to a certain degree, and goes to any situation where thecross-correlation between eigenstate is important. For a specific example,consider when the spatial probability density, i.e. the magnitude-squaredwave function, of a wave packet of excited eigenstates is of concern, similarto the case for our study in Chapter 5. If the eigenstate amplitudes maintainconstant magnitude, but are frequently disturbed in the phase values, thenwe can model cn = |cn|eiφn , where |cn| is deterministic but φn’s are inde-pendent, uniform random variables in the range [−pi, pi], with probability262.4. Numerical computationsdensity functions fn(φn) = 1/2pi. The statistically averaged result is then〈|Ψ(x, t)|2〉= 〈|∑n|cn|eiφne−iEnt~ψn(x)|2〉=∑n|cn|2|ψn(x)|2 +∑i 6=j|ci||cj |〈ei(φi−φj)〉e−i(Ej−Ei)tψi(x)ψ∗j (x)=∑n|cn|2|ψn(x)|2 (2.50)since〈ei(φi−φj)〉 =∫ pi−pi∫ pi−piei(φi−φj) 12pi12pidφidφj = 0. (2.51)Namely, the wave packet loses any time-dependent movement, originally dueto the coherent interference between eigenstates.Similar results apply to optical waves, and to the bichromatic coherentcontrol as well. In equations (2.43) (with ∆ω −∆ω′ = 0) and (2.44), if theoptical phases φ and φ′ are the random variables as described above, sta-tistical averaging would also make 〈cos(φ− φ′)〉 = 0, since 〈cos(φ− φ′)〉 =[〈ei(φ−φ′)〉+〈e−i(φ−φ′)〉]/2. Therefore, incoherence, exhibited in the random-ization of phases for eigenstates, would be expected to destroy any coherentwave packet motion, and coherent control abilities utilizing laser lights.2.4 Numerical computationsAll studies included in this thesis have some parts involving numerical com-putations. Since the material systems are based on diatomic molecules, weuse the atomic unit system in all numerical studies, where fundamental con-stants such as Bohr radius a0, electron mass and charge me and e, Planckconstant ~ and Coulomb constant 1/4pi0 become base units (numeric valuesof 1) for related physical quantities. Unit conversion into atomic units fromother unit systems can then be derived based on these base units. It’s alsoa common practice, in many places in this thesis, where “atomic unit” or“a.u.” is used to denote the unit of a numeric value; the underlying physical272.4. Numerical computationsquantity, e.g. energy or length1/2, is implied by the variable taking on thenumeric value.In solving for the numeric solutions to partial or ordinary differentialequations, functions of a spatial coordinate is represented by an array ofnumeric values (e.g. complex numbers), with predefined, equal spacing.For example, a wave function ψ(R) in the range of R ∈ [1, 11]a.u., withspatial resolution of 0.1a.u. is represented by a sequence of complex valuesψj = ψ(Rj), with Rj = (1.0 + 0.1j)a.u., j = 0...100. First and secondderivatives of the function are calculated using finite-difference, two-pointformuladψdR∣∣∣∣Rj= ψj+1 − ψj−12h ,d2ψdR2∣∣∣∣Rj= ψj+1 − 2ψj + ψj−1h2 (2.52)where h = Rj −Rj−1. End points use appropriate single-point formulas, orapproximated from neighbouring ones, and they are of less importance indetermining the solutions to the differential equations.To solve the next value in time also using finite-difference methods, eitherfor a discretized function as above, or simply a complex value, a number ofmethods ranging from Euler’s method to 4th order Runge-Kutta methodsare used. For any of these methods, the time increment is decreased untilconvergent results are obtained for a finite time propagation.Matrix diagonalization is also used when appropriate, such as solvingfor the eigenstates to a time-independent Hamiltonian matrix, representingthe kinetic energy and a known potential function −(1/2µ)d2/dR2 + V (R).According to equation (2.52), using the array of spatial points as basis, thematrix for (h2)d2/dR2 is tri-diagonal, with −2’s on the diagonal and 1’s leftand right to it, and that for V (R) is diagonal, with [V (R)]jj = V (Rj).The programming languages and visualization tools used for numericalstudies include FORTRAN, Python, C++, Mathematica and Gnuplot. Forelectronic structure calculations, GAUSSIAN quantum chemistry softwareis used.28Chapter 3Adiabatic photoassociationof ultracold molecules viascattering resonances3.1 IntroductionIn recent years, the production of translationally and internally cold (≤ 1K)and ultracold (≤ 1mK) molecules has become a very popular and intriguingsubject of research [6–11], due to the very novel and interesting physicsinvolving ultralow energy atomic collisions and Feshbach resonances [12–14], field-matter interactions [15–20], its relevance to ultracold chemistry [21,22], as well as potential applications these ultracold molecular systems canprovide, for example, for testing fundamental laws [23–26] or implementingquantum computers [27–31].Many methods to produce ultracold molecules exists. They can bebased on cooling previous hot molecules, such as buffer gas cooling [32],or Stark [33] and Zeeman [34–36] decelerators. Alternatively, many meth-ods rely on associating atoms already at ultracold temperatures into ultra-cold molecules, including, for example, one-colour photoassociation [37, 38],Feshbach resonance magneto-association [39–43], Feshbach-optimized pho-toassociation [44, 45], broadband femtosecond photoassociation [46–48], andmany more. These techniques have various advantages and disadvantages,in terms of technologies required, efficiencies, and constraints on propertiesof the material systems.Of particular interests to our current theoretical and numerical study is293.1. Introductionthe Adiabatic Raman Photoassociation (ARPA), introduced [49–51] as anextension of the Stimulated Raman Adiabatic Passage (STIRAP) [5, 52–55]to atomic scattering continuum states. This technique has unique char-acteristics from the dynamics of multicolour pulsed photoexcitation andphotoassociation, and meanwhile is also closely related to the mechanicsof scattering resonances. In this perspective, both static and dynamic as-pects of the photoexcitation and collision would play important roles in theprocess. Accordingly, our main inspiration and focus of the current studyis really the interesting dynamical phenomena as well as its effect on im-proving the efficiency of the technique. Comprehensive analysis for manyaspects on the practicality of the ARPA technique, or its comparison withexisting methods on producing ultracold molecules, calls for future researchbased on many references above.In the ARPA technique, as illustrated in Figure 3.1, the two-atom sys-tem initially consists of two separate freely colliding atoms, at ultralow tem-peratures, or equivalently ultralow collision energies. This is reflected inassuming that the initial quantum state of the system consists of contin-uum eigenstates in a very small energy range, just above the dissociationthreshold of the ground electronic potential energy surface with zero angularmomentum (J = 0 channel or the s-wave channel). Under the irradiation ofa combination of a dump and a pump laser pulse, the two-atom scatteringcontinuum is then coupled optically to single bound state in the excited elec-tronic manifold, which in turn is coupled to the absolute ground vibrationallevel of the ground electronic manifold. The use of full Born-Oppenheimerenergy eigenstates, also the small number of them, is ensured by the verynarrow energy bandwidth of the atomic collision, and the correspondingnarrow bandwidth of the optical excitation field.The pulse pair is mutually coherent and partially overlapping in spaceand time. Similar to the three-level STIRAP, the two pulses are ordered“counter-intuitively” [5]. The dump pulse precedes the pump (see section2.2.4), and has a central frequency appropriate for transition between the fi-nal bound state and an intermediate excited bound state. The pump pulse inturn has frequencies appropriate for coupling the continuum to the interme-303.1. IntroductionFigure 3.1: A schematic display of the Adiabatic Raman Photoassociation(ARPA) process. Left panel: Atoms colliding in the near-threshold energyrange are excited by the “pump” laser to the vibrational state |2〉 on anexcited electronic potential. The latter is coupled by the “dump” laser tothe deeply bound target state |1〉. Right panel: The same setup butwith the inclusion of a hyperfine-manifold bound level interacting with theprevious continuum, resulting in a Feshbach resonance.diate state. This way, an adiabatic passage is provided, optically connectingthe scattering continuum to deeply bound molecular states [49–51].The main limitation for the above approach to work efficiently is thesmall Franck-Condon factors, i.e. the spatial wave functions overlaps, be-tween the intermediate bound state and the ground potential. To overcomethis difficulty, Feshbach scattering resonances are then introduced, to pro-vide more spatially localized continuum wave functions, hence better over-lap. Figure 3.1 also illustrates a fundamental setup of such a Feshbach reso-nance: A different hyperfine electronic ground potential (green curve) withslightly higher dissociation threshold enables the interaction between one ofits high-lying vibrational level and the initial continuum states. Further, theenergy difference between the two ground potentials is magnetically tune-able, thus offering us a choice of the structure of the Feshbach resonance,particularly its width, based on which vibrational level is put into the energyrange.The use of wide resonances has been shown to be able to alleviate313.2. Dynamics of quantum-state amplitudesthe small-overlap limitation, in the Feshbach-optimized Photoassociation(FOPA) technique [44], which relies on the CW vibrational de-excitation ofheteronuclear diatomic molecules, and in the first single-collision analysis ofthe ARPA technique with resonance [56]. However, dynamic effects of theresonance become important, especially in the setting of pulsed photoexci-tation, for a collection of colliding atoms: Since the lifetime or duration ofthe atomic collision is inversely proportional to the width of the resonance,one can not overlook the increased number of atomic collisions in progress,hence available to enter the adiabatic passage, due to narrower resonances.A wide resonance, accordingly, shortens the collision time despite having animproved overlap factor. This becomes a key factor on choosing resonancewidth for best ultracold molecular production efficiency, when a gaseousensemble of atoms is considered.In the following, we first develop the underlying dynamical equationsthat describe the ARPA process. The effective mode expansion [57] methodis introduced, to aid to efficient numerical calculations for the ARPA pop-ulation dynamics, both for a single pair of ultracold 85Rb atoms (a singlecollision event), and for an atomic ensemble. There we demonstrate how thecollision lifetime is extended by narrow resonances, which can then leveragethe increased number of combining atoms during the photoexcitation, andprovide better overall efficiencies than wider resonances.3.2 Dynamics of quantum-state amplitudes3.2.1 Equations for bound-continuum population transferAs illustrated in Figure 3.1, the ARPA process involves the ground andfirst-excited electronic potential energy surfaces of the two-atom system.The initial state of the system consists entirely of energy eigenstates |E〉 inthe ground manifold, with energies above the dissociation threshold. Thedipole matrix element (transition amplitudes) are non-negligible betweenvibrational levels (one of which is |2〉) in the first excited electronic manifoldand initial continuum states, as well as vibrational levels in the ground323.2. Dynamics of quantum-state amplitudesmanifold, specifically, its lowest level |1〉.With the presence of a pair of optical pulses with (scalar) electric fieldE = n(t) cos(ωnt) (n = 1 for dump and n = 2 for pump), the systemundergoes transitions into both bound manifolds, provided that the two-atom continuum wave packet comes in close internuclear distances duringthe onset of the fields. The bandwidth of the pulses (determined by thedurations of n(t)) are chosen to match the ultracold temperature range ofthe atomic gas, centred around 100 µK and going down to tens of nK. Thisway, by choosing the appropriate centre frequencies ω1,2, we effectively needto include no other bound states than |1, 2〉, since the vibrational energyseparations in the electronic manifolds concerned are much larger.The effective Hamiltonian, therefore, can be written as (in atomic units),Hˆ = Hˆ0 − 2µˆ∑n=1,2n(t) cos(ωnt), (3.1)Hˆ0 = E1 |1〉 〈1|+ E2 |2〉 〈2|+∫ ∞EthE |E〉 〈E| dE, (3.2)where Eth is the dissociation (threshold) energy of the electronic manifoldhosting the continuum, and µˆ is the dipole operator of the molecule along theelectric fields’ direction. We assume, as usual, that the spatial variation (i.e.wavelength) of the optical field is much larger than the sizes of the atoms ormolecules, and hence is considered constant over spatial coordinates in theHamiltonian.The main feature of the atomic continuum we wish to explore is aFeshbach-type resonance, originating from the interaction between a high-lying vibrational bound level in one ground hyperfine electronic manifold,with the continuum portion of another hyperfine manifold with slightlyhigher dissociation threshold. We treat the two hyperfine manifolds togetheras one continuum (“the atomic continuum”) with each energy eigenstate |E〉accounting for bound and continuum states in both the closed and open hy-perfine channels. This, in the time domain, simplifies the collision dynamicsto only linear combination of continuum states |E〉 with time-dependentphase factors. It also leads to, in the energy domain, the expression of a333.2. Dynamics of quantum-state amplitudesresonance as the sharp energy dependence of the transition-dipole matrixelements µ2,E = 〈2| µˆ |E〉 [44, 56, 58].Expanding the total, time-dependent state of the molecule in terms ofthe material eigenstates involved,|Ψ(t)〉 =∑i=1,2bi(t)e−iEit |i〉+∫ ∞EthbE(t)e−iEt |E〉 dE, (3.3)and using their orthonormality and the Rotating Wave Approximation (RWA),the time-dependent Schro¨dinger’s equation i ddt |Ψ(t)〉 = Hˆ|Ψ(t)〉 can betranslated into a set of dynamical equations for the eigenstate amplitudesb˙1(t) = iΩ∗1(t)b2(t), (3.4)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + i∫ ∞EthΩE(t)bE(t)ei∆EtdE, (3.5)˙bE(t) = iΩ∗E(t)b2(t)e−i∆Et, (3.6)where ∆E = E2 − E − ω2 and ∆1 = E2 − E1 − ω1 are the energy detuningof the pulses, Ω1(t) = 1(t)µ2,1ei∆1t and ΩE(t) = 2(t)µ2,E are the Rabifrequencies, and Γf is a spontaneous decay rate we artificially added, atwhich amplitude of |2〉 is lost.We further reduce the number of dynamical equations by incorporatinga formal solution of equation (3.6)bE(t) = bE(0) + i∫ t0Ω∗E(t′)b2(t′)e−i∆Et′dt′, (3.7)into the integral of equation (3.5),b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + i∫ ∞EthΩE(t)bE(0)ei∆EtdE−2(t)∫ ∞−∞[|µ2,E |2∫ t02(t′)b2(t′)e−i∆Et′dt′]ei∆EtdE, (3.8)which can be further simplified by defining a source function and a spectral343.2. Dynamics of quantum-state amplitudesauto-correlation functionfsource(t) =∫ ∞EthΩE(t)bE(0)ei∆EtdE, (3.9)F (t− t′) =∫ ∞Eth|µ2,E |2ei∆E(t−t′)dE. (3.10)This makes the overall dynamical equations for the two bound statesb˙1(t) = iΩ∗1(t)b2(t), (3.11)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + ifsource(t)− 2(t)∫ t02(t′)b2(t′)F (t− t′)dt′.(3.12)Note that so far, these integro-differential equations are very difficult to solvenumerically. We will next further explore physical assumptions, especiallythose related to the inclusion of resonances, that can greatly reduce thecomplexity.3.2.2 The effective modes expansionIn the case of a flat continuum or “slowly varying continuum approxima-tion” (SVCA), µ2,E varies sufficiently slowly with energy E, and we re-place it by an averaged value without the energy dependence. This elim-inates the integral in the spectral auto-correlation function, reducing it toF (t− t′) = 2pi|µ2,E |2δ(t′− t). Consequently, the entire last term in equation3.12 depends only on b2(t), and the dynamical equation set becomes morecompact in matrix notationddtb = iH · b + ifsource, (3.13)353.2. Dynamics of quantum-state amplitudeswithb(t) ≡(b1(t)b2(t)), fsource(t) ≡(0fsource(t)),H =(0 Ω∗1Ω1 iΓeff (t)), Γeff (t) = pi|µ2,E |222(t). (3.14)A more detailed discussion of the solutions under SVCA was made in Refs.[49–51].The flat continuum or SVCA assumption becomes invalid with the pres-ence of collisional resonances, in which case µ2,E changes rapidly near theresonance energy [44, 58]. The effective mode expansion then starts by treat-ing µ2,E as an expansion of Lorentzian profiles [57]. Compact matrix formand efficient computation will then follow and greatly simplify the dynamicalequations. Specifically, let’s writeµ2,E =M∑s=1iµsΓs/2E − Es + iΓs/2, (3.15)where µs represents the electronic transition dipole moment, Γs the full-width-at-half-maximum (FWHM), and Es the centre energy for each reso-nance labelled by s. This form is capable of approximating well both wideand narrow resonances [56, 58, 59], meanwhile also evaluating the auto-correlation function integral analytically toF (t− t′) =M∑s=1αsf+s (t)f−s (t′), (3.16)withαs =∑s′−iµsµs′ΓsΓs′/4Es − Es′ − i(Γs + Γs′)/2, f±s (t) =√2pie∓iχst,χs = Es − E2 + ω2 − iΓs2 . (3.17)363.2. Dynamics of quantum-state amplitudesUsing equation (3.16) we now define the effective modes variables as [57],B−s (t) = i∫ t02(t′)b2(t′)f−s (t′)dt′, (3.18)which transforms the dynamical equations intob˙1(t) = iΩ∗1(t)b2(t),b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + ifsource(t) + i2(t)M∑s=1αsf+s (t)B−s (t),B˙−s (t) = i2(t)f−s (t)b2(t), s = 1, ...,M. (3.19)This way, we eliminated the integration in the original equation set at the(very limited) expense of increased number of equations. Further changingvariables Bs(t) =√αs/2pif+s (t)B−s (t) and Ω(s)2 ≡ 2(t)√2piαs, we haveb˙1(t) = iΩ∗1(t)b2(t)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + ifsource(t) +∑siΩ(s)2 (t)Bs(t)B˙s(t) = −iχsBs(t) + iΩ(s)2 (t)b2(t), (3.20)which can now again be put into matrix notation,ddtb = iH · b + ifsource, (3.21)whereb(t) =b1(t)b2(t)B1(t)B2(t)..., fsource(t) =0fsource(t)00..., (3.22)373.2. Dynamics of quantum-state amplitudesandH =0 Ω∗1 0 0 · · ·Ω1 iΓf Ω(1)2 Ω(2)2 · · ·0 Ω(1)2 −χ1 0 · · ·0 Ω(2)2 0 −χ2 · · ·............ . . .. (3.23)The effective modes amplitudes Bs(t) thus effectively describes somebound states of energies Es, coupled by the Rabi frequencies Ω(s)2 (t) to state|2〉, with detuning Es − E2 + ω2, and decay at rates Γs/2 as contained inχs. They reflect exactly the nature of bound-continuum interaction behindthe existence of scattering resonances. Note that the non-Hermiticity of theHamiltonian is due not just to the decay of the effective modes, appearing asthe imaginary part of χs, but also to αs in the Rabi frequencies Ω(s)2 , whichare in general complex numbers due to the summation in its definition.Equation (3.21) resembles a multi-state STIRAP [60] process with Ω1(t)and Ω(s)2 (t) coupling respectively |1〉 with |2〉, and |2〉 with each of the ef-fective modes (Figure 3.2). However, the transfer dynamics differs in asignificant way: in our current case the effective modes are initially empty,and get gradually populated. We can see this most explicitly for a singleresonance, for which the dynamical equations assume the form,b˙1(t) = iΩ∗1(t)b2(t)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + iΩ(1)2 (t)[fsource(t)/Ω(1)2 (t) +B1(t)]B˙1(t) = −iχ1B1(t) + iΩ(1)2 (t)b2(t). (3.24)383.2. Dynamics of quantum-state amplitudesFigure 3.2: Left panel: Three bound states STIRAP. The population whichstarts in the initial bound state |3〉 is transferred into state |1〉 by followingthe evolution of the “dark” field-dressed state. The process avoids popu-lation loss due to spontaneous emission from state |2〉 because the latterremains unpopulated. Middle panel: ARPA via a collisional resonance.The population gradually feeds the resonances as the continuum wave packet(shaded area) arrives (at time t0). Right panel: The resonance-dominatedcontinuum of the middle panel is replaced by a single (or several) effectivemode(s) with decaying amplitude(s) B1(t) (Bs(t)).Let B(t) = fsource(t)/Ω(1)2 (t) +B1(t), we obtainb˙1(t) = iΩ∗1(t)b2(t)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + iΩ(1)2 (t)B(t)B˙(t) = −iχ1B(t) + iΩ(1)2 (t)b2(t) +[iχ1fsource(t)/Ω(1)2 (t) + f ′source(t)].(3.25)where f ′source(t) ≡ i√2piα1∫∞−∞∆Eµ2,EbE(0)ei∆EtdE.Here, we can observe that this new effective mode variable B(t) hasdecaying behaviour at a rate indicated by the imaginary part of χ1. Thesource of population going into this effective mode comes from the termsin the square bracket (see Figure 3.2). There the first term is especiallyimportant, since it represents the field-free population source, as the Fouriertransform of the product of the resonant transition dipole moment and theenergy distribution of the initial scattering wave packet. The width of theresonance in comparison with the atomic scattering energy bandwidth will393.3. Numerical computation for Rb-Rb collisionsthen determine the characteristic time duration of this field-free source term.Such contrast with traditional bound-state STIRAP implies the possibil-ity of population loss from the “dark” state. In three-state counter-intuitivepulse ordering adiabatic passage [5], the population resides initially in theadiabatic “dark” state, which is a superposition of the initial and targetstates only. The adiabaticity of the pulses guarantees the completeness ofthe transfer from the initial to the target state, leaving the intermediatestate unpopulated at all times. Since, in our case, the effective mode getspopulated in a gradual fashion, the system wave function may contain non-negligible contributions from other “bright” states that have a small overlapwith |2〉, allowing population loss via spontaneous emission. The way tocounter the effect of increased spontaneous emission at higher atomic gastemperatures is discussed in section 3.3.3.3.3 Numerical computation for Rb-Rb collisionsIn this section we employ the above formulation and perform some numericalcomputations on the resonance-enhanced photoassociation of ultracold 85Rbatoms to form 85Rb2 in its ground vibrational state. In keeping with our viewof the process, we divide the computations into two parts: the populationtransfer for a single collision event involving one pair of atoms, and that fora thermal gaseous ensemble of atoms.3.3.1 Single-event transfer efficiencyFollowing the model of Refs. [50, 51], we consider a pair of 85Rb atomscolliding on the (J = 0) ground electronic potential. We assume that att = 0 (before the onset of the pulses) the total molecular state wave packethas non-zero amplitudes only in the continuum eigenstates|Ψ(0)〉 =∫ ∞EthbE(0)e−iEt |E〉 dE, (3.26)403.3. Numerical computation for Rb-Rb collisionsFigure 3.3: A single resonance is used for the computation, with magnitude|µ2,E | = |µresΓres/ [2(E − Eres) + iΓres]| as a function of energy, shown forvarious choices of Γres and Eres = 100µK. The resonance height is µres =300a.u., but can be made even larger to favour lower laser intensities. TheFranck-Condon factor involving eigenstate |E〉 follows the same shape.and the continuum wave packet is an energetically-narrow Gaussian [49–51]bE(0) =1(2piδ20)1/4exp[− (E − E0)22δ20+ i(E − E0)t0], (3.27)with δ0 = 70µK and E0 = 100µK (temperature is converted to energy inatomic units with E = kBT ). The threshold energy Eth will from nowon be taken as −∞ for effective evaluation of the integrals, based on theassumption that bE(0) has essentially decayed to negligible values muchbefore the true Eth. With this choice of parameters, the wave packet’stemporal peak occurs at a numeric value t0 = 1.2µs [49, 50]. Of course, inpractice timing the excitation pulses with respect to a single collision eventis unrealistic. However, since we will calculate in the next section the totalphotoassociation efficiency for an ensemble of atoms colliding at differenttimes, the specification of such timing is important. Lastly, in order for thepulse pair’s spectral widths to have a good overlap with the energetic spreadof the atomic wave packet, the pulses in the time domain have µs durations.The scattering continuum is assumed to contain a single resonance,whose shape is given by equation (3.15). Figure 3.3 shows this resonancecentred at Eres = E0 = 100µK for three different width choices. Since Eres413.3. Numerical computation for Rb-Rb collisionscan be tuned in practice (e.g. magnetically), we can optimize the transferby making it always equal to E0. This is confirmed by by the numericalcalculation (Figure 3.5 bottom panel) to improve the maximal overlap en-hancement.The bound-to-bound matrix element is chosen to have a numeric valueµ21 = 0.0051a.u. This numeric value can be different depending on the ac-tual experimental set-up, but our results only depend on the Rabi frequencyΩ1(t), which is proportional to the product of this bound-to-bound matrixelement with the dump pulse amplitude. A increase (or decrease) of the ma-trix element translates into a proportional decrease (or increase) in the laseramplitude required. The spontaneous decay rate from |2〉 is Γf = (30ns)−1[49–51]. The centre frequency of the dump pulse is chosen to coincide withE2 − E1, and that of the pump pulse with E2 − E0. The field envelopes1,2(t) are taken as Gaussian functions, peaking respectively at 1.05µs and1.55µs. The duration of both fields is 0.22µs, peaking at intensity 3 × 105W/cm2 [50].With these specifications, Eqs. (3.20) becomeb˙1(t) = iΩ∗1(t)b2(t), (3.28)b˙2(t) = iΩ1(t)b1(t)− Γfb2(t) + ifsource(t) + iΩ(1)2 (t)B1(t), (3.29)B˙1(t) = −(Γres/2)B1(t) + iΩ(1)2 (t)b2(t), (3.30)where Ω1(t) = 1(t)µ2,1 and Ω(1)2 = 2(t)µres√2piΓres/2. Notice here that theFranck-Condon factors contained in µ2,1 and µres always appear as productwith the field amplitudes 1,2(t). So the intensities of the fields really need tobe chosen according to the Franck-Condon factors between states involved.An enhancement of either one of them will result in the same order decreaserequired for the other.Figure 3.4 shows the results of numerically propagating the equationsfor b1,2(t) and B1(t), given that b1(0) = b2(0) = B1(0) = 0. We plot thepopulations of states |1, 2〉 for a wide (100µK) and a narrow (6µK) resonance.The wide resonance results in an essentially complete population transfer(> 90%), while that of the narrow one is only ∼ 23%. No loss of population423.3. Numerical computation for Rb-Rb collisionsFigure 3.4: The target |1〉 and intermediate |2〉 state populations as a func-tion of time for two resonance widths. Top panel: the pump and dumpfield amplitudes. Middle panel: The target state population |b1(t)|2. Thetransfer efficiency is 90% for the wide (100µK) resonance, but only 23% forthe narrow (6µK) resonance. Bottom panel: The intermediate state pop-ulation |b2(t)|2. (Notice the large difference in the vertical scale relative tothe middle panel.)433.3. Numerical computation for Rb-Rb collisionsFigure 3.5: Top panel: The target population, |b1(t → ∞)|2, for differentresonance widths. Bottom panel: |b1(t→∞)|2 as a function of the centreof the resonance Eres, for E0 = 100µK; this shows transfer is optimal whenthe centre of the resonance coincides with the central energy of bE(0).in the target ground state can occur in a single atom-pair photoassociationevent, but it is possible in practice, due to either atomic collisions thatpopulate other vibrational levels, or multiple pulse pairs that subsequentlymove population back or away. As shown in the lower panel, due to theadiabatic nature of the process and the “counter-intuitive” pulse ordering,the population of the intermediate level |2〉 remains very low.3.3.2 Ensemble transfer efficiencyIn agreement with Ref. [56], for each single atom-pair photoassociationevent, the transfer efficiency via a wide resonances is always better than thatof a narrow one (Figure 3.5 top panel). The situation is however different443.3. Numerical computation for Rb-Rb collisionsFigure 3.6: The magnitude of the window functions fW (t) for differentresonance widths Γres and fixed height µres. Longer tails of |fW (t)| areobserved for narrower resonances.for an ensemble of colliding atoms due to previously mentioned dynamicaleffects. An alternative way of viewing this effect is to examine the discreteeffective modes which replace the continuum in our theory. These modesare, to all intents and purposes, resonances[61]. As clearly seen in equation(3.30), the rate of de-populating an effective mode is Γres, the resonance-width of that mode. Hence narrower resonances correspond to smaller ratesof depopulation, and increased interaction duration of the effective modeswith |2〉.In Figure 3.6 we examine these trends in a quantitative way by displayingfW (t), the field-normalized source term (or a window function), given as,fW (t) = fsource(t)/2(t) =∫ ∞−∞µ2,EbE(0)ei∆EtdE, (3.31)for resonances of changing widths. Clearly in evidence is the prolongedduration of fW (t) when switching to narrower resonances.The temporally stretched population source is also beneficial when weconsider the action of a pulse pair that is delayed relative to t0, the arrivaltime of the incoming wave packet. Figure 3.7 shows the transfer efficiency asa function of such delay times for 3 different resonance widths. For a narrowresonance, despite the drop in the peak value, the single collision transferefficiency remains large for longer times. This means that atom-pairs that453.3. Numerical computation for Rb-Rb collisionsstarted their collision prior to the peak fields can still be transferred intobound molecules with significant probability.Figure 3.7: Target state transfer efficiency |b1(t → ∞)|2 for different reso-nance widths as a function of the delay time δt ≡ t0 − tP , where tP = 1.2µsis the pulses’ overlap peak time, and t0 is the incoming wave packet peaktime.Figure 3.8: The time-averaged molecular production efficiency for a singlepulse pair for an atom ensemble at 100µK, calculated by integrating thedelay plots of Figure 3.7.In order to obtain the delay-time averaged efficiency for an atomic en-semble, we need to calculate the area under the transfer-efficiency curvesin Figure 3.7. Figure 3.8 displays the dependence of the delay0time aver-aged efficiency for various resonance widths. We first notice that the ef-ficiency changes relatively slowly for resonance width larger than 1000µK.This is because in this case the resonance width far exceeds δ0, the energetic463.3. Numerical computation for Rb-Rb collisionsspread of the initial atomic ensemble, thus approaching the flat-continuum(SVCA) limit. But as the width of the resonance drops to a few µK, themolecular production efficiency rises to a maximum value, then drops signif-icantly due to prolonged collision time making spontaneous emission morepronounced. Therefore, there really exists an optimal resonance width forwhich the molecular production efficiency is maximal. Comparing the op-timal molecular production efficiency, obtained for a (narrow) resonancevalue of ∼ 8µK, with the efficiency in the flat-continuum limit, we see animprovement factor around 1.56.3.3.3 Scaling behaviour with ensemble temperatureWe now explore, as was done in Ref. [49], how the process varies as theaverage ensemble energy E0 and energy spread δ0 are scaled down by afactor of s > 1, namely, E0 → E0/s, and δ0 → δ0/s.The equations have been shown to be invariant under this scaling [49],provided that the peak time was scaled up by the same factor t0 → st0, andthe initial wave packet amplitude is scaled as bE(0;E, t0)→√sbE (0;E/s, st0).We now consider the effect, in addition to the above, of scaling the resonanceshape as µ2,E(E,Γres)→ µ2,E (E/s,Γres/s). In order to match the spectralprofile to the scaled bE(0), we need to scale up the centre frequencies anddurations of the two pulses by the same s factor. Since we can choosethe intensities of the fields, we let 1(t) → 1(ts)/s and 2(t) → 2(ts)/√s[49] As a result, the source function scales like fsource(t, t0,Γres) → s−1 ·fsource(st, st0,Γres/s). The above scaling choices leave the dynamical equa-tions 3.28-3.30 essentially unchanged, only except for the spontaneous decayrate. As we scale the relevant times by a factor of s, the effect of the spon-taneous emission becomes more pronounced.In Figure 3.9, we display the dependence of the single collision photoas-sociation efficiency on the spontaneous decay rate. Note that a change inone order of magnitude for the spontaneous decay rate only affects our re-sults negligibly. When we compare the results to those in Figure 3.5, wherethe transfer efficiency is plotted as a function of the resonance width, we see473.3. Numerical computation for Rb-Rb collisionsFigure 3.9: Top panel: The single collision photoassociation efficiency|b1(t → ∞)|2 as a function of resonance width, for four different valuesof the spontaneous decay rate. Bottom panel: The same plot at strongerlaser intensities. The photoassociation efficiency now becomes insensitive tothe spontaneous decay rates.483.3. Numerical computation for Rb-Rb collisionsthat the transfer efficiency is not significantly affected at ensemble temper-atures of a few µK to a few 102µK. As the ensemble temperature goes downby three orders of magnitude, the single collision transfer efficiency goesdown too, by ∼ 67% of its original value. The effect is more pronouncedfor narrower resonances, because the longer interaction times enhance theeffect of the spontaneous decay. However, as shown in the lower panel ofFigure 3.9, it is possible to counter the effect of spontaneous decay at verylow temperatures, e.g. in nK range, by increasing the intensity of both laserfields.So in short, the optimal resonance width seems to be in most casesaround ∼ 8% of the ensemble temperature. The corresponding molecularproduction efficiency is ∼ 56% higher than that of the wide resonance (flatcontinuum) case.3.3.4 Thermalization and practical mattersSo far, we have considered the molecular production efficiency for a singlecollision event, and for an ensemble of colliding atoms, under the excitationof a single pulse pair. In practice, one might also need to consider theeffect of a train of such pulse pairs, carrying out the photoassociation on thesame atomic ensemble. Although this is not the focus our current study, wenevertheless provide some qualitative comments on the difficulties and theirpossible solutions.Firstly, the cumulative action of a train of pulse, at longer timescales,can in fact change regions in phase space corresponding to the recombiningatoms, thereby affecting the initial wave packet amplitude bE(0). However,an atomic ensemble can thermalize sufficiently fast, on the order of millisec-onds, to yield the typical atomic trap setting of 100µK temperature and1011/cm3 density [62]. This means, depending on the repetition rate of thepulses, that after a few thousand µs pulse-pairs, the atomic ensemble canthermalize back to its original phase-space distribution, re-validating theensemble-averaged form we used for bE(0).According to previous estimates [49, 50], the total number of pulse pairs493.4. Conclusionsneeded to transfer an entire atomic ensemble of density 1011/cm3 (at atemperature of 100 µK) is around of 107. Therefore a few thousand pulsesis indeed a very small fraction of total number of pulses needed, and thethermalization is fast enough comparing with the ensemble size molecularconversion time.Another practical difficulty is how to hide the newly formed molecule, ei-ther from subsequent pulse pairs that may transfer them back to the atomiccontinuum, or from surrounding atomic collisions that can impart temper-ature to the ultracold molecules. In accordance with more detailed discus-sions in Ref [49, 50], this can be done, for example, by allowing the newlyformed molecules to physically “leak” away from the laser focus, which ispossible because they react differently from the atoms to the confining laserfrequency. A molecular trap can then be placed just below the atomic trap.Such solution, however, imposes constraints on the time separation betweenpulse pairs in the pulse train; their separation needs to be long enoughfor such molecule-atom separation process to happen. Further quantitativeanalysis, as future extension to this study, is then required.3.4 ConclusionsIn this study we have shown that Adiabatic Raman Photoassociation ofultracold atoms, assisted by collisional resonances, is an efficient way of pro-ducing ultracold diatomic molecules in deeply bound states. By replacingthe atomic scattering continuum with resonances by a discrete set of ef-fective modes, we derived an analytical framework for the dynamics thatboth reveals the factors contributing to the final molecular production ef-ficiency clearly and concisely, and makes convenient computational imple-mentations. Then through numerical simulations, we confirmed the improve-ment of molecular production efficiency, in the single atom-pair collisionalevent, using wide resonances. When an ensemble of atoms is considered, onthe other hand, there exists an optimal, yet very narrow, value of resonancewidth for best transfer efficiency. The underlying physical mechanism is thedynamic interplay and balanced competition between large Franck-Condon503.4. Conclusionsoverlap factors, which improve single collision transfer efficiency, and pro-longed collisional lifetimes, which increase the number of collision eventsduring the optical pulses.Quantitatively, the narrow-resonance molecular production efficiency canbe as much as ∼ 56% higher than the wide resonances efficiency. For atomictemperatures in the µK range, we find that the optimal conditions are at-tained for resonances whose widths are about one order of magnitude smallerthan that of the ensemble temperature. In the case of lowered atomic tem-perature, ARPA scheme is shown to also work efficiently. The only non-scalable, or non-controllable, factor is the spontaneous decay rate, which inthe decreased temperature setting, amplifies its harm to the final produc-tion efficiency. It is shown, however, that it can be effectively countered byincreasing intensities of the optical fields used.Interesting future applications would include time-dependent resonances.We envision combining ARPA with dynamical sweeping Feshbach resonancesacross the threshold energy range. As the sweep will render the resonancesnarrower, the laser pulses will be made narrower in order to transfer theatomic gas into molecules in an optimal piecewise manner.51Chapter 4Linear molecular response inthe strong field domainIn this chapter, we discuss the dynamical behaviour of quantum transi-tions of simple molecules under the excitation by broadband, high-intensity,femtosecond pulses that are subject to coherent spectral shaping. The in-teresting physics resides in the interference effects in the photo-response ofthe molecule, specifically how the transition into a number of previously un-occupied quantum states can be correlated and controlled by the coherentaspects of the incident light pulse.The original motivation behind the studies is the following. In time-dependent quantum mechanics, the probability for a material system totransition to a new energy eigenstate of the unperturbed Hamiltonian, dueto a first-order perturbation (see Section 2.2.2), is linearly dependent on thestrength of such perturbation [1]. However, in practice, the strength of theperturbing force is usually difficult to classify in relation to the quantitiesof the system it applies to. Therefore, the linearity in the material responsehas been commonly used, although incorrectly, to deduce whether the per-turbation is of the first order. It is then one of our motivations to clarifythis fallacy by demonstrating that such linearity is not exclusive to first or-der perturbative fields: linear response can also be present with strong field(non-first order) due to dynamical interference effects.In addition, some recent experiments during the time of the original pub-lication also claimed phase-only coherent control using weak optical fields[63] - an impossibility according to traditional coherent control theory (sec-tion 2.2.3 and [4, 64, 65]). Although, later on, these experiments stimu-524.1. Linear response via Raman transitions in a three-potential modelPRLNuclear coordinateEnergy-10-5 0 5 10 15 20 0  0.5  1  1.5  2  2.5  3  3.5  4-10-5 0 5 10 15 20 0.6  0.8  1  1.2  1.4  1.6  1.8  2Thursday, 3 November, 11Figure 4.1: The model molecular system has two energy-degenerate excitedelectronic potentials of continuum nature, coupled with a number of boundvibrational levels in the ground potential by a broadband light pulse.lated the even richer topic of coherent control over open quantum systems[66–69], where such phase-only control is shown possible, they did employabove incorrect linear-response argument to produce its conclusions aboutthe strength of the light fields. We therefore consider our studies also asa possible alternative explanation to these experimental findings, by pre-senting response linearity, high-intensity perturbation, as well as phase-onlycoherent control in the same picture.4.1 Linear response via Raman transitions in athree-potential modelThe first study below presents a rather general formulation, describing theresponse of a molecular quantum system under a strong optical perturbation.We will reveal the linear response of the material system under high-intensityexcitation signified by higher order excitation, accompanied by coherent con-trol. Specifically, the molecular system initially resides in a bound manifoldalong a certain internal coordinate (e.g. vibration of a diatomic molecule),and the incident optical pulse interacts with the molecule’s dipole, and ex-cites it electronically into a pair of energy-degenerate excited potentials thathost continuum or quasi-continuum states (e.g. for dissociation or isomer-ization), as illustrated in Figure 4.1.534.1. Linear response via Raman transitions in a three-potential modelWith the electric field of the pulse E(t) = 2ˆ(t) cos(ωLt) being polarizedin ˆ direction, having envelope modulation (t) and centre frequency ωL, theeffective Hamiltonian in atomic units isH = HM +HI ,HM =N∑j=1Ej |Ej〉 〈Ej |+∑n∫E |E,n−〉 〈E,n−| dE,HI = −µ ·E(t), (4.1)where µ is dipole operator of the molecule, |E,n−〉 are the aforementionedcontinuum eigenstates of energy eigenvalue E, and n denotes one amongthe two potentials (the “channel”) which is also the main quantity we wishto control. N is the number of bound states that are effectively coupledby (continuum-mediated) resonance Raman transitions as enabled by thebandwidth of the excitation pulse.As usual, the time-dependent state of the material system is expandedin relevant energy eigenstates,|Ψ(t)〉 =N∑j=1bj(t)e−iEjt |Ej〉+∑n∫bE,n(t)e−iEt |E,n−〉 dE. (4.2)The Schro¨dinger’s equation for the time evolution of this state then trans-lates into the following differential equations (with Rotating Wave Approx-imation)b˙(t) = i∑n∫ΩE,n(t)bE,n(t)dE,b˙E,n(t) = iΩ†E,n(t) · b(t), (4.3)where we have organized the bound level amplitudes and Rabi frequencies asvectors [b(t)]j ≡ bj(t) and [ΩE,n(t)]j ≡ (t)µj,n(E)e−i∆E,jt. The transitiondipole matrix elements are µj,n(E) ≡ 〈Ej |µ·ˆ |E,n−〉, and the laser detuningfrom resonance is ∆E,j = E − Ej − ωL.The time evolution of the bound state amplitudes is obtained by formally544.1. Linear response via Raman transitions in a three-potential modelintegrating bE,n(t), and substituting the expression into the first part ofequation (4.3), assuming that bE,n(−∞) = 0 for all E and n:b˙j(t) = −(t)eiEjt∫ t−∞∗(t′)∑k,nF (n)j,k (t− t′)bk(t′)e−iEkt′dt′, (4.4)whereF (n)j,k (t− t′) =∫µj,n(E)µn,k(E)e−i(E−ωL)(t−t′)dE . (4.5)The matrix spectral correlation function F (n)j,k (t−t′) can be further simplifiedby applying the “slowly varying continuum approximation” (SVCA) [4], inwhich we replace each energy-dependent transition dipole matrix element bysome average value µj,n(E) ∼= µj,n(ωL) ≡ µ¯j,n, and it follows that F (n)j,k (t −t′) ∼= 2piµ¯j,nµ¯n,kδ(t − t′). This approximation is valid whenever the bound-continuum transition dipole matrix element change little over the energiescovered by the pulse’s bandwidth. The result is a simplified set of equationsfor the bound amplitudesb˙(t) = −ΩI(t)b(t), (4.6)[ΩI(t)]j,k = pi|(t)|2∑nµ¯j,nµ¯n,kei(Ej−Ek)t, (4.7)which can be readily propagated numerically. With bound coefficients known,imposing bE,n(−∞) = 0, we have the probability amplitude of excitation (or,photodissociation or photoisomerization) to channel n asPE,n = |bE,n(t→∞)|2,bE,n(t) = i∫ t−∞Ω†E,n(t′) · b(t)dt′. (4.8)We now assume that two bound vibrational states |E1〉 and |E2〉 in theground potential are coupled to two continuum channels n = α, β, andcoherent control is then reflected in our ability to change the energy-averagedbranching ratioρα,β =∫ ∞−∞PE,αdE/∫ ∞−∞PE,βdE. (4.9)554.1. Linear response via Raman transitions in a three-potential modelThe generality of our formulation lies partly in the way that we can definedimensionless parameters qn = µ¯2,n/µ¯1,n, and Q = µ¯1,β/µ¯1,α. Although qα,βand Q are complex in general, we simplify the analysis by choosing themto be real. We note that when qα = qβ, writing out equation (4.8) wouldgive us the branching ratio as ρα,β = 1/Q2, which is independent of thefield, thus no optical control is possible in this case. Therefore, we assumeqα 6= qβ, a very reasonable assumption in most molecular systems.In a typical experimental setup, a pulse shaper [70] is used to specify theamplitudes and phases of each frequency component of the excitation pulse.Here we consider a class of pulses consisting of two equally wide, linearlychirped sub-pulses. The interaction Hamiltonian that follows has generalformHI = −2µ[(t+ t0) + (t− t0)] cos[ω(t)t]∼= −µs(t)e−iωLt,ω(t) = ωL + a0t,s(t) ≡[(t+ t0) + (t− t0)]e−ia0t2 . (4.10)For our analysis, we have devised a specific pulse form, as the following,which includes a pair of Gaussian-shaped sub-pulsess(t) = I1/2[2(1 + 2iA)(1 + e−t20/τ2)]−1/2·{exp[ −(t+ t0)22τ2(1 + 2iA)]+ exp[ −(t− t0)22τ2(1 + 2iA)]}(4.11)which results in a convenient, analytical expression for the spectrum˜s(ω) =1√2pi∫ ∞−∞s(t)e−iωtdt= cos (ωt0)(2I)1/2τ exp[iφ/2− iAτ2ω2 − ω2τ2/2][1 + cosφ exp(−t20/τ2)]1/2 . (4.12)The purpose of such analytical construction is to guarantee the conserva-tion of total pulse energy, for all dimensionless parameters related to shaping564.1. Linear response via Raman transitions in a three-potential model 0 0.2 0.4 0.6 0.8 1-20-16-12-8-4  0  4  8 12 16 20Populationtime [/o]pulse shapebound level 1bound level 2continuum 1 totalcontinuum 2 total 0 0.2 0.4 0.6 0.8 1-20-16-12-8-4  0  4  8 12 16 20Populationtime [/o]pulse shapebound level 1bound level 2continuum 1 totalcontinuum 2 total 0 0.2 0.4 0.6 0.8 1-20-16-12-8-4  0  4  8 12 16 20Populationtime [/o]pulse shapebound level 1bound level 2continuum 1 totalcontinuum 2 total 0 0.2 0.4 0.6 0.8 1-20 -16 -12 -8 -4  0  4  8  12  16  20Populationtime [/o]T0=0,A=0 0 0.2 0.4 0.6 0.8 1-20 -16 -12 -8 -4  0  4  8  12  16  20Populationtime [/o]T0=2/,A=0 0 0.2 0.4 0.6 0.8 1-20 -16 -12 -8 -4  0  4  8  12  16  20Populationtime [/o]T0=2/,A=/ 0 0.2 0.4 0.6 0.8 1-20 -16 -12 -8 -4  0  4  8  12  16  20Populationtime [/o]T0=0,A=/ 0 0.2 0.4 0.6 0.8 1-20-16- -8-4  4 8 12 16 20Populationtime [/o]pulse shape (T0=0,A=0)bound state 1bound state 2continuum _continuum `Figure 4.2: Sample photo-products population dynamics for different timedelays and chirping parameters. The intensity value is µ2τI = 0.01.574.1. Linear response via Raman transitions in a three-potential modeldefined as the following. Firstly, W = τ |E1−E2| is used to represent the du-ration for both sub-pulses, in the unit of the inverse of the energy separationbetween the vibrational states. The time delay between the two sub-pulses isthen controlled by the value of T0 ≡ t0/τ , where 2t0 the is the separation intime between the peaks of the sub-pulses. Lastly, A determines any poten-tial linear chirp rate via a0 = A/[τ2(1+4A2)]. Such parameterization allowsseparate control of the spectral amplitude via T0 and the spectral phase viaA. Note that the choice (T0 = 0, A = 0) corresponds to a transform-limitedpulse with temporal profile s(t) = I1/2 exp[−t2/(2τ2)].Given the pulse shape, the probabilities of photodissociation (photoiso-merization) into the n = α, β channels are calculated using equation (4.8).For the case where we use Q = 2, qα = 0.6, qβ = 1.2 and W = 0.5, Figure4.2 shows the probabilities as functions of time. Because of the assumed“flatness” of the continuum, the changes in the probability are rather mono-tonic in time. Also, having started with state |E1〉 , state |E2〉 is populatedby continuum-mediated resonance Raman transitions. Figure 4.3 displaysP (I), the photoexcitation yield given as the total population removed fromthe bound state manifold after the pulse is over, with P (I) =∑n PE,n, wherePE,n are obtained from equation (4.8). We see that the response curves, insubplots (a,c), start saturating after displaying a linear dependence. Atmoderately strong intensities, (substantial) control over the branching ra-tio, in subplots (b,d), as a function of T0 and A is clearly in evidence. Forexample, at the higher end of intensity values (µ2τI = 0.01−0.02), the yieldfor the (T0 = 0, A = 0) pulse in subplot (a) displays a short bout of linearI dependence followed by the onset of saturation. In contrast, the yield forthe (T0 = pi,A = 0) case in subplot (a), or (T0 = 0, A = pi) cases in subplot(c), displays a much more protracted linear phase prior to saturation.In spite of the nearly linear photoexcitation yield, for example, at inten-sity values around µ2τI = 0.01, the branching ratios exhibit rather extensiverange of control, of about 16% for shaped pulses (middle non-flat branchingratio curves, in second and fourth panels in Figure 4.3). Even at aroundµ2τI = 0.005 (lowest non-flat branching ratio curves), where all the yieldcurves are essentially the same and highly linear, the range of branching584.1. Linear response via Raman transitions in a three-potential model 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`T0 µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IT0=0 or 2.0/; A=0.00/A=0.25/A=0.50/A=0.75/A=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`AT0=0; µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IA=0; T0=0.00/T0=0.25/T0=0.50/T0=0.75/T0=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`T0 µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IT0=0 or 2.0/; A=0.00/A=0.25/A=0.50/A=0.75/A=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`AT0=0; µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IA=0; T0=0.00/T0=0.25/T0=0.50/T0=0.75/T0=1.00/ 0 0.2 0.4 0.6 0.8 1 0  0.005  0.01  0.015  0.02Excited State Populationµ2 o IA=0, T0= 0.00/T0= 0.25/T0= 0.50/T0= 0.75/T0= 1.00/  . . . . 1  .005  0.01  0.015  0.02Excited State Population2 o IT0=0,2.0/, A=0.00/A= 0.25/A= 0.50/A= 0.75/A= 1.00/(a)(b)(c)(d) 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`T0 µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IT0=0 or 2.0/; A=0.00/A=0.25/A=0.50/A=0.75/A=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330. 0/ 0.25/ 0.50/ 0.75/ 1.00/l_,`AT0=0; µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IA=0; T0=0.00/T0=0.25/T0=0.50/T0=0.75/T0=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330.00/ 0.25/ 0.50/ 0.75/ 1.00/l_,`T0 µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IT0=0 or 2.0/; A=0.00/A=0.25/A=0.50/A=0.75/A=1.00/ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.330. 0/ 0.25/ 0.50/ 0.75/ 1.00/l_,`AT0=0; µ2 o I = 0.0200µ2 o I = 0.0150µ2 o I = 0.0100µ2 o I = 0.0050µ2 o I = 0.0001 0 0.2 0.4 0.6 0.8 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o IA=0; T0=0.00/T0=0.25/T0=0.50/T0=0.75/T0=1.00/ 0 0.2 0.4 0.6 0.8 1 0  0.005  0.01  0.015  0.02Excited State Populationµ2 o IA=0,  .00T0= 0.25/T0= 0.50/T0= 0.75/T0= 1.00/  . . . . 1  .005  0.01  0.015  0.02Excited State Population o IT0=0,2.0/  A=0.00/A= 0.25/A= 0.50/A= 0.75/A= 1.00/(a)(b)(c)(d)Figure 4.3: The photoexcitation yields or “response curves” (subplots a,c) asfunctions of the field intensity, at particular shaping parameter choices, andthe branching ratios as functions of shaping parameters, at fixed intensities(subplots b,d). The response curves are made more linear due to shaping.594.1. Linear response via Raman transitions in a three-potential model 0 0.2 0.4 0.6 0.8 0  0.005  0.01  0.015  0.02Nonlinearityµ2 o I0T0=0, A=0A= 0.25/A= 0.50/A= 0.75/A= 1.00/ 0 0.2 0.4 0.6 0.8 0  0.005  0.01  0.015  0.02Nonlinearityµ2 o I0A=0, T0= 0T0= 0.25/T0= 0.50/T0= 0.75/T0= 1.00/Figure 4.4: The nonlinearity of the intensity response curves vs. µ2τI0.ratio control is still about 8%, obviously a non first-order behaviour. Onlywhen the intensity approaches zero (e.g. at µ2τI = 0.0001) where the yieldapproaches zero too, does the variation in the branching ratio due to shapingdisappear.In order to quantify the visibly evident high degree of linear intensitydependence, we have also calculated a nonlinearity factor(d2PdI2∣∣∣∣I0I0)/(2dPdI∣∣∣∣I0),the ratio between the second-order and the first-order terms in the Taylorseries expansion about I0. Figure 4.4 shows the degree of nonlinearity calcu-lated in this manner, plotted against µ2τI0. We see that until µ2τI0 = 0.02,where all the curves begin to saturate significantly, pulses whose shapes aredefined by (T0 = pi,A = 0) or (T0 = 0, A = pi) exhibit a greater degree oflinearity, as compared to transform-limited pulse (i.e. (T0 = 0, A = 0)). Upto µ2τI0 = 0.01, the nonlinear factors of the above pulses are about halftheir transform-limited values.604.1. Linear response via Raman transitions in a three-potential modelThe physical reason for this enhanced linearity of the molecular responseto pulse shaping is due to the coherent interplay between the two sub-pulses.Essentially, the first sub-pulse creates, by a (continuum-mediated) Ramanprocess, a superposition of the |E1〉 and |E2〉 bound states. The secondsub-pulse then dissociates this superposition state in the usual bichromaticcontrol scenario [4, 64]. We verify this mechanism by looking at the in-tensity responses, a subset of which is shown in subplot (a) of Figure 4.3,as a function of T0. For each T0 value, when it’s displaced by an integermultiple of 2pi (as a result of our choice of numerical parameters), we foundan identical response curve. This matches the periodicity of the vibrationalsuperposition. However, when we set qα and qβ to zero, the variations inT0 no longer alter the intensity response curves, nor do such variations af-fect the branching ratio. For cases involving chirping, the response curvescan only mainly a certain degree of “recurrence” for A smaller than 2pi, atT0  1.Finally, we also examine here the case in which the initial state iscomposed of a coherent superposition of the two bound states, c1 |E1〉 +eiφ0c2 |E2〉 with c21 + c22 = 1, subject to the action of a simple transform-limited excitation pulse, for which (T0 = 0, A = 0). This time, it is thevariation in c2, or the relative phase φ0, that affects the intensity responsecurves. In Figure 4.5, we see changes to linearity of the curves, now due tochange in the c2 and φ0 parameters. The variation in the branching ratio,due to changes in these variables, is quite significant. The branching ra-tios are however insensitive to the laser intensity because control is mainlyderived from the variation in the initial superposition rather than the field.In summary, here we have been able to probe how pulse shaping affectsthe intensity dependence of the photoexcitation, by first developing a time-dependent, non-perturbative model of coherent control of a three-potentialmolecular system interacting with a broadband pulse. We have shown thatshaped pulses may exhibit a wide range of linear response of the overallphotoexcitation yield, even at moderately high intensities. The mechanismresponsible for this behaviour is bichromatic control in conjunction withcontinuum-mediated Raman transitions between bound states. Secondly,614.1. Linear response via Raman transitions in a three-potential model 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0  0.1  0.2  0.3  0.4  0.5l_,`c2µ2oI = 0.0200µ2oI = 0.0100µ2oI = 0.0050µ2oI = 0.0001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  1  2  3  4  5  6  7l_,`q0µ2oI = 0.0200µ2oI = 0.0100µ2oI = 0.0050µ2oI = 0.0001 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Ic22=0.0c22=0.1c22=0.2c22=0.3c22=0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Iq0=0.00/q0=0.25/q0=0.50/q0=0.75/q0=1.00/ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Iq0=0.0q0=0.2q0=0.5q0=0.7q0=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Iq0=0.00/q0=0.25/q0=0.50/q0=0.75/q0=1.00/ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Iq0=0.00/q0=0.25/0 .50 .70 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0  0.02  0.04  0.06  0.08  0.1Excited State Populationµ2 o Iq0=0.00/q0=0.25/q0=0.50/q0=0.75/q0=1.00/ . . . . . . . . .   .  .  .Excited State Population  IFigure 4.5: The intensity response curves for an initial superposition of twobound states ψi = c1 |E1〉+ eiφ0c2 |E2〉. Upper two panels are the intensityresponse and control for φ0 = 0 and variable c2; lower two panels are theresponse and control at fixed c22 = 0.4 and variable φ0.624.2. Linear response in H+2we have shown that extensive coherent control over the branching ratiocan co-exist with extended linear intensity regimes. Thus, linear intensityresponse, which we have shown to occur even for moderately strong pulseswhere non-perturbative effects are present, is not a necessary indicator forthe weak-field regime nor the validity of employing first order perturbationtheory.4.2 Linear response in H+2As we have understood from the above section, there are cases in which theresponse of a molecule in a single eigenstate to the effects, especially coher-ent ones, of a high-intensity, broadband laser field is linear. The underlyingmechanism can be attributed to quantum interferences between multiple-eigenstate excitation pathways. In this second study, we demonstrate theextended appearance of such response linearity in the realistic hydrogencation molecule, H+2 , that undergoes extensive rotational and electronic ex-citation by a femtosecond pulse. This study relates to and extends theprevious one in two aspects:(i) The quantum structure of H+2 , although being the “simplest molecule”,is more complex and realistic than the three-potential model before. Molec-ular rotation now provides a “ladder” of degenerate potential energy sur-faces within each electronic manifold. We tackle the problem accordingly bysimulating the dynamics of vibrational wave packets, rather than full Born-Oppenheimer energy eigenstate amplitudes, in these rotational-electronicchannels. It turns out to be conceptually compact, analytically concise,computationally straightforward and non-demanding, and meanwhile alsoreveals the much richer wave packet interference effects.(ii) The initial state of the molecule is assumed to be a vibrational wavepacket in one particular rotational-electronic channel, as a continuation fromthe last case analyzed in the previous study. Such wave packet is assumedto be created from a single vibrational eigenstate by some earlier pulse viaRaman-type transitions. As such, it’s important to point out that in our cur-rent case, we always have bichromatic control even with first-order pertur-634.2. Linear response in H+2bative field due to the simultaneous excitation of multiple vibrational eigen-states. Nevertheless, we will consider and demonstrate non-perturbativeregime exclusively. The excitation pulse used is thus high-intensity andbroadband, and its timing with respect to the initial wave packet can betuned. Various intensity response and population dynamics behaviour willthen follow, in every relevant rotational-electronic channel, as a result ofsuch time delay.The time-delayed optically probing of wave packet interference effects iscommonly known as a Wave Packet Interferometry (WPI) technique [4, 64,71–75]. The WPI in our current setup differs from a conventional one in thatthe photoexcitation is delayed with respect to a pre-existing material wavepacket, rather than a precursor pulse. The resultant wave packet interferenceis nevertheless similar: Due to the high intensity of the excitation pulse, theinitial vibrational wave packet interferes with part of itself, via Raman-type transitions, differently at different delay times. Also, the wave packetscreated by the excitation pulse originating from each eigenstate within theinitial wave packet also interfere differently at different delay times, and doso in an extended number of bound and continuum channels.As before, not only do we reveal linear response (now as a result ofWPI-type excitation), we will also show the coexisting ability of coherentcontrol. It also represents a simple yet effective control mechanism: the timedelay translates, in the spectral domain, to a linear tilt of the spectral phaseof the excitation pulse. Although such linear shaping may not affect thetwo-photon spectrum of the pulse, it does so for higher order multiphotonspectral components.4.2.1 Vibrational wave packet methodThe molecule under consideration, H+2 , interacts with an incident coherentlaser pulse with centre wavelength at 304nm (32900 cm−1), chosen to overlapwell energetically with the transition frequencies between the ground 1sσgelectronic state and the first-excited 2pσu electronic state of the molecule(Figure 4.6). We denote the H−H+ inter-nuclear separation vector by R,644.2. Linear response in H+2-0.10-0.050.000.050.10 1  2  3  4  5  6  7  8Potential Energy (Hartree)Internuclear distance (Bohr)1smg2pmuFigure 4.6: Schematic illustration for the photoexcitation of H+2 by an opticalelectric field with bandwidth covering three vibrational states of the ground1sσg state.the collective electronic coordinate by r, and the electronic wave function byΦα(R, r), where α = g, e refers to the ground or excited electronic state. Thevibrational wave packet approach starts by expanding the total molecularwave function in the YMJ (θ, φ)Φα(R, r) rotational-electronic basis,Ψ(t,R, r) =∑α,J,Mψα,J,M (t, R)YMJ (θ, φ)Φα(R, r)/R, (4.13)where J and M denote respectively the quantum number for total angu-lar momentum of the nuclei and its z-projection. Instead of vibrationaleigenstates, time dependent “vibronic coefficients” ψα,J,M (t, R) are used torepresent the vibrational wave packet in the channel specified by (α, J,M),and thus depend on time non-trivially. The additional choice of linear fieldpolarization ensures that ∆M = 0, and that ∆J = ±1.Dropping the conserved M index for brevity, we consider the moleculestarting in the J = 0,M = 0 ground electronic state g. The molecule’squantum dynamics is governed by the time dependent Schro¨dinger equationi~∂Ψ∂t = HΨ, (4.14)654.2. Linear response in H+2where as usual H is the total light-matter dipole interaction HamiltonianH = Hmol. − µ ·E(t) = Hmol. − µ · EˆE(t), (4.15)with µ being the dipole operator and EˆE(t) the electric field of the excitationlight. Substituting the state expansion and the Hamiltonian into equation(4.14), we thus obtain a set of coupled differential equations in matrix formi~ ddtψ(t, R) = H(t, R)ψ(t, R). (4.16)where ψ(t, R) is represented by a vector of vibrational wave packets, definedto have components ψJ = ψe,J for odd J , and ψJ = ψg,J for even J , withJ = 0, 1, 2, 3... The Hamiltonian matrix H is a tridiagonal matrix, withHJ.J = He,J for odd J , and HJ,J = Hg,J for even J , with diagonal elementsbeingH(g,e),J(R) = −~22md2dR2 +J(J + 1)2mR2 +Wg,e(R), (4.17)and off-diagonal elementsHJ,J+1 = HJ+1,J = −〈Y 0J+1| 〈Φe|µ |Φg〉 ·E(t) |Y 0J 〉= −D(R)E(t) 〈Y 0J+1| cos θ |Y 0J 〉 , (4.18)where m is the reduced mass for the molecule, and θ is the angle between Eˆand the transition dipole momentum 〈Φe|µ |Φg〉, a spatial vector assumedto be along the internuclear axis direction. The ground and excited poten-tial energy surfaces Wg,e(R), and the magnitude of the electronic transitiondipole function D(R), accurate for R ≤ 17 Bohr, are calculated using GAUS-SIAN with TD method and 6-311+G(3df,3pd) basis set. Using propertiesof the associated Legendre polynomials, we also have simplification〈YMJ+1| cos θ |YMJ 〉 =√(J + 1 +M)(J + 1−M)(2J + 1)(2J + 3) . (4.19)In the dipole interaction, the electric field amplitude E(t) is spatially664.2. Linear response in H+2uniform on the molecular scale, and has only time dependence. We choosea transform-limited pulse,E(t) = E0 exp[−(t− tc)2/2t2p]cos [ωL(t− tc)] , (4.20)with peak intensity I = c0|E0|2. The pulse duration is chosen to be tp = 3fs, so that its width in energy (∼ ~/tp) is comparable to the differences inenergy between the v = 4, 5, 6 vibrational levels that make up the initialwave packet in the J = 0 channel (Figure 4.6),ψg,J=0(t = 0, R) =[φ4(R) + φ5(R) + φ6(R)]/√3ψα,J>0(t = 0, R) = 0, (4.21)created at some initial time t = 0. Here φv(R) are the formal vibrationaleigenstates of the channel-Hamiltonian Hg,J=0(R).We then apply the laser pulse of equation (4.20) and follow the wavepacket dynamics, governed by the Schro¨dinger’s equation, using a 4th or-der Runge-Kutta numerical method, with 0.0006 fs time steps and a spatialgrid mesh of 0.02 Bohr. With such short pulse duration, accurate numer-ical propagation of vibrational wave packets can be achieved at moderatecomputer times. An advanced numerical method utilizing effective modeexpansion [76] provides an efficient, full-eigenstate alternative to our vibra-tional wave packet method, especially in the case when the excitation pulsehas much longer duration.4.2.2 Response analysisThe propagation of vibrational wave packets enables us to directly obtainthe molecular population in each specific channels they belong, as a functionof the intensity of the excitation field. In Figures 4.7 to 4.10, we display themolecular populations in rotational channels up to J = 10, for a wide rangeof peak intensity values. The populations are obtained by integrating themagnitude-squared vibrational wave packets over the internuclear distanceR. The upper bound for rotation J = 10 is chosen to be appropriate for the674.2. Linear response in H+2peak intensity of the field, which is verified by gradually increasing the peakintensity until the excited population in J = 10 channel is approaching, butkept below, 1%.As shown in Figure 4.7 and 4.8, for 9 < tc < 20 fs range of delay timeswe observe a variable degree of near linear response of the population to thefield intensity. This is observed for all the rotational channels, as well as inthe initially unoccupied v = 3, 7 vibrational states. The response becomesmore linear for example at tc ∼= 18 fs, when suppression of excitation, suchas in the J = 0 → J = 2, 4 case discussed below, exists. An additionalcase of essentially perfect linear response occurs for tc ∼= 18 fs where thev = 3 state exhibits excitation enhancement. Strong linearity accompaniedby excitation enhancement is also observed in v = 7 state and J = 1, 3channels, at around tc ∼= 9 fs. At relatively high rotational channels J > 5,response can still be very linear close to tc ∼= 15 fs, but this occurs only atmoderate to high intensities (> 200TW/cm2).Such essentially linear response curves can persist up to a few hundredTW/cm2 of peak intensity. This is despite the fact that at such intensi-ties the fields display all the features of being strong: For example, startingat the ground rotational level, the field induces extensive rotational excita-tion by multiple Raman transitions, populating channels up to J = 10. Inaddition, coherent control over the branching ratios between rotational, orro-vibrational, channels, is also present as a function of tc alone (as shownin Figure 4.11). This amounts to a part, or a second stage, of the phase-only (referring to the molecular phases determined by tc) coherent controlover the excitation of a single eigenstate [4], that, again, is only present inthe non-perturbative regime. We have thus conclusively demonstrated ouroriginal claim [77] that essentially linear response curves can in fact existwithin the strong-field regimes, accompanied by phase-only coherent controlover the population branching ratio.In Figures 4.8 and 4.10, we display the response of the system whenthe delay times assume values displaced by ∼ 20 fs relative to the values ofFigures 4.7 and 4.9. They are almost identical, hence implying a periodicbehaviour. This periodic behaviour is best demonstrated in Figure 4.12 and684.2. Linear response in H+24.13, where clear population beating, at about the same period ∼ 20 fs, isdisplayed in all the rotational and vibrational states under consideration.The period of ∼ 20 fs matches that of the vibrational motion of the initialwave packet, which is determined by the energy separation between nearestvibrational levels. Specifically, this period is given as 2pi~/∆E ∼= 2pi~/0.008Hartree ∼= 18.8 fs, with ∆E being the average energy difference between thev = 4, 5 and v = 5, 6 levels.A similar beating phenomenon to the one described here was observedexperimentally by Ohmori et al. [78], (see also [79]) where a strong delayednear-infrared excitation pulse, analogous to our second excitation pulse, wasseen to cause beatings in the populations of individual vibrational levelsbelonging to the B-state of diatomic iodine, due to the interference betweenthe elastic |v〉 → |v′〉 → |v〉 Rayleigh scattering and the inelastic |v ± 1〉 →|v′〉 → |v〉 Raman scattering.The present oscillatory behaviour is similar to the above mechanism[78], except for the important difference that the intermediate state is the|E,n−〉 dissociative continuum. We are thus seeing interference between theelastic |v〉 → |E,n−〉 → |v〉 resonance Rayleigh scattering and the inelastic|v ± 1〉 → |E,n−〉 → |v〉 resonance Raman scattering.Aside from such periodicity, the population beating in Figure 4.12 and4.13 is also influenced by the initial vibrational wave packet motion. InFigure 4.14 we plot the spatio-temporal profile of the initial field-free wavepacket: ∣∣∣φ4(R)e−iE4t/~ + φ5(R)e−iE5t/~ + φ6(R)e−iE6t/~∣∣∣2.We observe that at tc = 10, 30, and 50 fs delay times for which the populationof the entire J = 0 channel is minimal, the field-free wave packet reaches itsclassical turning point, and is mostly concentrated at the Franck-Condonregion close to R = 3 − 4 Bohr. This in turn increases the Franck-Condonoverlap and thus enhances the total population removal from the entireJ = 0 channel.694.2. Linear response in H+2Figure 4.7: State populations as a function of laser peak intensity, and ofdelay times for various J-channels, and for the v = 3 and v = 7 states ofthe J = 0 channel.704.2. Linear response in H+2Figure 4.8: State populations plots similar to Figure 4.7, but for a differentband of delay times approximately 20 fs later. The response curves showalmost identical behaviour.714.2. Linear response in H+2Figure 4.9: Similar state population plots as in Figure 4.7 and 4.8, butfor channels of higher J , with the same bands of delay times (same colourschemes for the plots).724.2. Linear response in H+2Figure 4.10: Continued from Figure 4.9, for the later time band.734.2. Linear response in H+2Figure 4.11: Branching ratio of populations in some pairs of vibrational androtational states are plotted, signifying coherent control.744.2. Linear response in H+2Figure 4.12: Population in various vibrational eigenstates v = 3 − 7 in theJ = 0 channel, and first three rotational channels, as functions of delay timetc. Superimposed are such scans with different field intensities, with thesame intensity range and colour scheme as Figure 4.11.754.2. Linear response in H+2Figure 4.13: Similar to Figure 4.12 but for higher rotational channels.764.2. Linear response in H+2Figure 4.14: The field-free probability-density for the initial vibrational wavepacket as a function of R and t. The beat frequency (≈ 5 × 1013 Hz) is inagreement with the average vibrational spacing (Ev=6 − Ev=4)/(2~).4.2.3 ConclusionsWe have studied the regimes of linear response of the H+2 molecule subject toa high-intensity ultrafast pulse, which is delayed with respect to an assumedprecursor Raman-type excitation that creates a vibrational superpositionstate. We have demonstrated that the photoexcitation yield as a function ofthe pulse’s intensity can vary greatly in shape, and exhibit essentially lineardependence at certain delay times. At moderately strong laser intensities,the molecular population in a number of rotational-electronic channels alsooscillate with this delay time, with a period corresponding to the vibrationalperiod determined by the nearest vibrational eigenstates. Strong-field linearresponse is linked to the suppression of excitation during the destructiveinterference periods of this oscillatory behaviour, where excitation saturationis “postponed” to even larger field intensities. This effectively makes thelinearity of the response curves an ineffective indicator of the strength ofthe field, comparing with, for example, the extent of rotational or electronicexcitation.This study complements the first study on a three-state model, and af-firms our previous findings concerning phase-shaped excitation pulses [77]:While the existence of weak fields can justify the use of first-order pertur-774.2. Linear response in H+2bation theory and imply linear response, the converse is not true. Namely,linear response does not necessarily imply the perturbative weak field do-main.It’s also important to point out that, in the context of high-intensityexcitation of the H+2 molecule, many other interesting phenomena such asnon-adiabatic wave packet dynamics around light-induced conical intersec-tions [80–84] are not considered in this study, mainly due to the very shorttime-scale of the photoexcitation considered. For strong, broadband pho-toexcitations with much longer durations, the molecular wave packet cre-ated via multiphoton transitions, would be able to traverse the field-dressedtwo-dimensional potential energy surface in both the bond length and rota-tional angle degrees of freedom, with a conical intersection formed due tothe ground and excited electronic manifolds intercepting in energy. Our cur-rent method, which accounts for the coupling between the vibrational androtational wave packet motion, would then in principle be able to analyzesuch non-adiabatic wave packet dynamics.78Chapter 5Quantum dynamics underincoherent photoexcitation5.1 Electronic decoherence via random vibration5.1.1 MotivationIn previous studies, we have focused on methods of manipulating and con-trolling molecular quantum dynamics through coherent properties of theoptical fields, e.g. their spectral phases and timing, in relation to the co-herent quantum mechanical wave packets in molecules. Crucial to achievingthe final control is our ability to affect the coherent aspects of the field,which in practice is enabled by coherent light sources such as lasers, andexperimental tools for spectral manipulation such as pulse shapers. Thus,the molecular control is in its nature inseparable from the coherence of theexcitation light.In recent years, an increasing number of studies have identified surpris-ingly long-lived coherences in biological systems, especially in photosynthesiclight harvesting molecules [85–93]. Such long-lived coherences were detectedin studies in two-color photon echo spectroscopy [86, 88], angle-resolved co-herent optical wave-mixing [87], and phase-stabilized 2D electronic spec-troscopy [94]. Following these findings, it has been suggested that quantumcoherence may play an important role in biological processes, for examplein the energy transfer that follows the photo-excitation step. An importantquestion to ask is, then, to what extent the dynamics observed in such ex-periments, which utilize coherent laser light sources, are relevant for naturalprocesses induced by incoherent light sources such as sunlight, even when the795.1. Electronic decoherence via random vibrationtwo types of light share the same central frequency and spectral bandwidth.It is the objective of this study to demonstrate the loss of quantum co-herence and control, found in setups similar to our previous studies or thelight-harvesting experiments, when the molecular system is excited by in-coherent light sources. Several previous studies, analyzing various effectsof incoherent excitation sources on the resultant molecular dynamics, in-clude an analysis of the excited state survival probability as a function ofthe incoherence of the light [95, 96], effects of incoherent fields on the pho-toisomerization yields [97], a quantum-optical formulation of the state ofthe molecule after photon-absorption [98], studies of open system dynam-ics relevant to photosynthetic complexes [99], and general features of theresponse of open systems to incoherent excitation [100]. However, none ofthese studies looked directly at the spatial shape of the wave packets andthe associated temporal evolution and coherence properties resulting fromexcitation with incoherent vs. coherent light sources. Such studies are par-ticularly important in comparing the nature of the excited quantum stateprepared by these light sources.Therefore, we will concentrate on the actual spatio-temporal shape ofthe wave functions resulting from photo-excitation, a primary entity thateffectively brings out the differences between the two types of light-inducedexcitations mentioned above. Our main analytical tool is a time-dependentquantum-theoretic calculation of the different nuclear wave packets result-ing from the photo-excitation step. Optical incoherence is then simulatedby introducing random jumps in the phase and central frequency of the ex-citation pulse, with the degree of optical incoherence being controlled by thefrequency and amplitudes of these random jumps.We will consider first the photo-excitation of a model molecule into a su-perposition of two electronic states, and contrast the calculated coherencesinduced by incoherent light with the outcome of excitation by an ultrashortcoherent pulse with the same central frequency and spectral bandwidth.The spatial structure and time dependence of the resultant wave packet dy-namics is then analyzed for a variety of potential energy surfaces. Followingit, we will present the similar random wave packet dynamics in the real-805.1. Electronic decoherence via random vibrationistic IBr molecule, in which the energy-degenerate electronic states are ofbound-continuum nature. In this case, the photodissociation flux will alsobe analyzed for the coherent vs. incoherent cases.5.1.2 Stochastic Schro¨dinger’s equation for random wavepacketIn this section, we develop a time-dependent theory describing the dynamicsof vibrational wave packets of a molecule under weak-intensity optical fieldthat may be incoherent.Consider, first, the case in which the molecule resides in the groundelectronic manifold with (J = 0,M = 0), and is photo-excited to a singleelectronic excited state that can be of dissociative or bound nature. Thelight source is again treated as a classical time-varying electric field. Thetime-evolution of the quantum state of the molecule can then be describedby a stochastic Schro¨dinger’s equation,i~ ddtΨ(t,R) = H(t)Ψ(t,R), (5.1)where the state vector is written in the basis of rotational-electronic channelsasΨ(t,R) = 1R(ψg(t, R)ψx(t, R)),with ψg,x(t, R) denoting the vibrational wave packets in the ground andexcited electronic manifold. The total Hamiltonian matrix consists of atime-independent molecular part and a time-dependent interaction part de-scribing electric-dipole interactionH(t) = Hmol + HI(t),Hmol =(Hg 00 Hx)=(TˆN +Wg 00 TˆN +Wx),HI(t) =(0 V ∗(t)V (t) 0). (5.2)815.1. Electronic decoherence via random vibrationThe nuclear kinetic energy operator TˆN , and the ground and excited PESfunctions Wg,x, are by definition functions of the set of nuclear coordinatesR. The matrix element describing interaction has a general form ofV (R, t) = −[∫Φx(R, qe)µ(R, qe)Φg(R, qe)dqe]· ˆE(t), (5.3)where Φg,x are electronic eigenstate wave functions involving collective elec-tronic coordinates qe, E(t) = EL(t) cos(ωt+ φ) is the electric field, µ(R, qe)is the dipole operator, and ˆ is the unit polarization vector.It is most convenient to solve the Schro¨dinger equation in the interactionrepresentation (with the R variable temporarily suppressed for brevity)i~ ddt[U†(t− t0)Ψ(t0)]= H′I(t)[U†(t− t0)Ψ(t0)], (5.4)where t0 represents some initial time long before the onset of HI(t), andH′I(t) ≡ U†(t− t0)HI(t)U(t− t0),with the evolution operator in the interaction representation defined asU(tf − ti) ≡ e−iHmol(tf−ti)/~ ≡(Ug(tf − ti) 00 Ux(tf − ti)). (5.5)Assuming the laser field intensity is weak enough so that first-order per-turbation theory is valid, the state in the interaction representation assumesthe formU†(t− t0)Ψ(t) = Ψ(t0) +1i~∫ tt0H′I(t′)Ψ(t0)dt′, (5.6)or in the Schro¨dinger’s representation,Ψ(t) = U(t− t0)[1 + 1i~∫ tt0H′I(t)dt′]Ψ(t0) (5.7)In order to gain physical insight, we now consider excitation by a E(τ)δ(t−825.1. Electronic decoherence via random vibrationτ) pulse, for whichV (R, t) = D(R)E(τ)δ(t− τ),D(R) ≡ −ˆ ·∫Φx(R, qe)µ(R, qe)Φg(R, qe)dqe. (5.8)The assumption that the molecule is initially in the ground electronic man-ifold,Ψ(t0,R) =1R(ψg(t0, R)0),reduces equation (5.7) toψx(t, R)/R =h(t− τ)i~ Ux(t− τ)D(R)E(τ)Ug(τ − t0)[ψg(t0, R)/R], (5.9)whereh(t− τ) =1 for t > τ12 for t = τ0 for t < τ(5.10)is the Heaviside function. The physical picture that emerges is that nopopulation is excited before the arrival of the delta pulse, and exactly att = τ , where h(0) = 1/2, and Ue(τ − τ) = 1, we haveψx(τ,R)/R =D(R)2i~ E(τ)Ug(τ − t0)[ψg(t0, R)/R]= D(R)2i~ E(τ)ψg(τ,R)/R. (5.11)Thus, at t = τ , a replica of the ground state wave packet (multiplied withD(R)E(τ)/2i~) is formed on the excited PES, and evolves at t > τ asψx(t > τ,R)/R = Ux(t− τ)[ψx(τ,R)/R]. (5.12)Using the above insight, we can consider any time-dependent functionfor the electric field, as a series of delta functions with the strength equal to835.1. Electronic decoherence via random vibrationthe field at that time pointE(t) =∫ ∞−∞E(τ)δ(t− τ)dτ. (5.13)This generalizes equation (5.9) for an arbitrary pulse, to obtainψx(t, R)/R =∫ t−∞Ux(t− τ)i~ D(R)E(τ)Ug(τ − t0)[ψg(t0, R)/R]dτ. (5.14)The excited wave packet can then be viewed, equivalently, as a sum ofreplicas launched by all delta pulses until time t, evolving according to theexcited state Hamiltonian Hx [101, 102]. The differential form of equation(5.14),i~ ddtψx(t, R)R = Hxψx(t, R)R +D(R)E(t)Ug(t− t0)ψg(t0, R)R , (5.15)is exactly an inhomogeneous Schro¨dinger’s equation for the excited statewave packet ψx(t, R), “driven” by a source term which is the product ofthe dipole strength along the polarization direction, the replication of theground electronic state wave packet at the given time, and a potentiallystochastic (incoherent) electric field.Equation (5.15) not only reveals the very direct relationship between theexcited wave packet and the electric field of the light, therefore also theircoherence, but also has the great advantage of allowing us to easily vary thedegrees of incoherence in the light sources. In particular, we can convenientlyintroduce random jumps in the central frequency and the carrier-envelopephase of the field, at any given time point. This is equivalent to doing sofor a selected number of the δ function components of the field in equation(5.13), with magnitudeE(τ) = EL(τ) cos{[ωL + ∆ω(τ)]τ + φ(τ)}. (5.16)The real-valued ∆ω(τ) and φ(τ), which are taken to be constant (zero) intime for coherent fields, become random variables for incoherent light. An845.1. Electronic decoherence via random vibration-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11- --1e-110e+001e-112e-113 -4  5  100  150  200  250Electric Field (a.u.)i  ftTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  1  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10- e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-1110e+01e-112e-113 14e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLi c r t(i)(ii)(iii)(iv)Figure 5 1: A realization of the incoherent light, and its magnified view,with centra wavelength at ∼455nm (0.1 Hartree), and envelope functionEL t) ∼ [sin(pit/2 0 fs)]0.1 for 0 < t < 200 fs, and zero elsewhere, chosen tosimulate CW behaviour without “sudden” numerical turning on and off. Arandom phase jump in the [−pi, pi] range, and a central frequency shift inthe ±0.0175 Hartrees range, is introduced every 7 fs on average.example is shown in Figures 5.1 for such a field suffering phase and frequencyinterruptions, with intervals averaging about 7 fs. The field correlation func-tion 〈E(τ + t)E(t)〉 is tested to fall off rapidly, becoming essentially zero onthe order of a few femtoseconds. In comparison, the coherence time of solarradiation is 1.32 fs [103]. However, such incoherent field, although con-venient to construct, is considered more coherent than true solar radiation,due the very flat and well defined envelope function. Therefore, this reducedand very elementary model of i coheren light can be thought of as ratherdescribing a single, atomic light emitter, e.g. from a thermal body, thatsuffers from relatively regular collisions that induce the phase and frequencyinterruptions [104]. In contrast, true solar light originates from thermal ra-855.1. Electronic decoherence via random vibration-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11- --1e-110e+001e-112e-113 -4  5  100  150  200  250Electric Field (a.u.)i  ftTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  1  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10-1e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-11-1e-110e+001e-112e-113e-114e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLincoherent-3e-11-2e-11-1e-110e+001e-112e-113e-11 0  10  20  30  40  50Electric Field  Magnitude (a.u.)Time (fs) 0  0.05  0.1  0.15  0.2Spectral Magnitude  (arbitrary unit)Energy (Hartree)coherent incoherent-3e-10-2e-10- e-100e+001e-102e-103e-10 0  3  6  9  12  15Electric Field (a.u.)Time (fs)incoherentcoherent-4e-11-3e-11-2e-1110e+01e-112e-113 14e-11 0  50  100  150  200  250Electric Field (a.u.)Time (fs)tTLTLi c r t(i)(ii)(iii)(iv)Figure 5.2: The coherent pulse has transform-limited envelope functionEL(t) ∼ exp[−(t− 6 fs)2/2(1.6 fs)2]. A portion of the incoherent pulseis also shown for comparison. The spectra of the pulses of coherent and asingle realization of incoherent light are also shown. The total energy flux,∫∞−∞ |E(t)|2dt, is kept the same for the two types of pulses.diation from a large number of emitters, each of which is expected to havefield properties similar to our current one. On such basis, we would expectany results following true solar radiation would only be an amplified versionof our conclusions.A coherent pulse is constructed from a spectrum matching that of theincoherent one, in central frequency and bandwidth, as shown in Figure 5.2.In the time domain, the coherent field is seen to be much shorter in durationthan the incoherent one, but with much larger peak amplitude, in order tocarry the same energy flux.The method adopted below is similar in spirit to the “quantum trajec-tories” approach[105, 106] where the ensemble associated with the electricfield correlation function is built up of a collection of individual realizations,865.1. Electronic decoherence via random vibrationeach one given by excitation with different random parameters. Two differ-ent types of results are shown: those from sampling individual realizations,and those obtained as an average over the ensemble of realizations.5.1.3 Wave packet simulations and electronic decoherenceTo conduct a detailed numerical simulation for the quantum wave packetdynamics under the photoexcitation descried in Figure 5.1 and 5.2, we con-sider a model molecular system, built from the parameters from the PESof the H+2 molecular ion. The initial wave packet is assumed to be in theground 1σg electronic state with angular momentum J = 0, and is photoex-cited to two model excited states that can each be dissociative or bound.We also assume that these two excited states are decoupled from one an-other, in order to reduce the dynamics to that of a sum of the individualground-to-excited-state transitions, each described by equation (5.15).At t0 = 0, we assume for simplicity that the ground state wave packetis a vibrational eigenstate, making its time-evolution trivial, and that theexcitation field is linearly polarized, making the transition from (J = 0,M =0) only to (J = 1,M = 0). The transition dipole function is modelledaccording to H+2 [107], as D(R) = D(R) ≈ R/2 in atomic units. Equation(5.15) then becomesddtψx(R, t)R = −iHx(R)ψx(R, t)R +D(R)E(t)e−iEgtψg(R)R . (5.17)withHx(R) =−12µ( d2dR2 +2RddR)+ 1µR2 +Wx(R), (5.18)where µ is the molecule’s reduced mass, and Wx(R) is the potential energysurface for the excited state, which will take various shapes for differentcases in the following. Now the 1/R factorization conveniently transformsthe above equation to a one-dimensional Schro¨dinger’s equationddtψx(R, t) = −i[−12µd2dR2 +Wx(R)]ψx(R, t) +D(R)E(t)e−iEgtψg(R),(5.19)875.1. Electronic decoherence via random vibrationwhich is ready for numerical computation.Using the same electric field E(t) as in Figures 5.1 and 5.2, and choosingψg(R) as the v = 5 vibrational state, we used equation (5.19) to propagatethe excited radial wave functions ψ(1)x (R, t) and ψ(2)x (R, t) in electronic states1 and 2, from t = 0 to 300 fs, using time steps of 0.003 fs and radial stepsof 0.02 Bohrs. We analyze the motion of the ψ(1)x (R, t) and ψ(2)x (R, t) pairthat has originated from the same ground electronic state for two cases: (A)Continuum dynamics for a pair of purely repulsive exponentially-decayingexcited potentials, and (B) Bound state dynamics for a pair of Morse po-tentials, where a heavier reduced mass, µ = 5 × 918 a.u. is used. Sampleplots of the wave packets, for these two cases, subject to the coherent andincoherent modes of excitation, are presented in Figures 5.3 and 5.4. In thecase of the incoherent excitation, the results of a single random realizationis shown.The completed wave packets resulting from incoherent light excitationare seen to be choppy and highly unstructured. In the case of the repul-sive potentials, following the excitation step, these wave packets spread outmuch more rapidly, while covering much larger extensions, than do the co-herently excited cases. The situation is even more dramatic in the boundstate excitation cases where the wave packets resulting from the incoherentlight excitation exhibit a significantly smaller number of coherent oscillationsas compared to the excitation by coherent pulses, and a highly irregularlystructured wave function is seen.The above results show characteristics of the excited state wave functionson individual electronic states. Also of interest, in particular with respectto experimental studies of long-lived coherences, is the persistence of elec-tronic coherence, a property related to several electronic levels. To examinethis, we quantify the electronic coherence by the off-diagonal element of thedensity matrix of the two electronic states, traced over the nuclear spatialcoordinate,ρ1,2(t) =∫[ψ(1)x (R, t)]∗ψ(2)x (R, t)dR. (5.20)Because the coherence is obtained via first-order perturbation theory, it is885.1. Electronic decoherence via random vibrationwith TL pulse with incoherent pulse401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolute-valu -squared of excited nuclear wave packets as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and state-2. Upper panel - the wave packets excited by chaotic light, in state-1 (upper row) andstate-2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the total trace of the density matrix. Inthis way we factor out the e↵ects of the total excitationyield that are of no interest for our present comparativestudy.Labelling the two excited electronic states as state-1and state-2, the electronic coherence assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions in both electronic statesare assumed to have the same angular momentum (J =1,M = 0), the above TrR operation amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two types of radi-ation, we also considered the e↵ects of the PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the shape of the PES of state-2. We do so by ei-ther translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Figure 3 displays the choices of the PES for thetwo electronic states, as well as the coherence under theaction of the two types of fields.These plots clearly show that for chaotic field exci-tations the magnitude of the coherence ⇢(t) decays ata rate much faster than that of the ultrafast coherencepulse, approaching a much lower asymptotic value thanthat of the transform-limited case. This is especiallyso, as shown in the middle row of Figure 3, when thePES asymptotic energy separation is comparable to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsooccurs when the Franck Condon (FC) regions of the twosurfaces are significantly di↵erent. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-limited one. Thus we find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figure 3) the twoPES have the same asymptotic energies, though the ini-tial decay of the coherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long times. This is understandable since at large in-ternuclear distances, when the electronic states becomedegenerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to point out that decoherence here isnot due to environmental e↵ects because our molecule isisolated from an outside environment. In the chaotic lightcase the decoherence is a result of the random nature ofthe two wave packets and their moving away from oneanother. If, in addition, coupling to the environment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tronic coherence in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by calculating electronic coherences in the pho-todissociation of a molecule into a superposition of twoelectronic states, by examining two types of electromag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolute-value-squared of excited nuclear wave ck ts as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and state-2. Upper panel - the wave packe s excited by chaotic light, in stat -1 (upper row) andstate-2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the total trace of the density matrix. Inthis way we factor out the e↵ects of the total excitationyield that are of no interest for our present comparativestudy.Labelling the two excited electronic states as state-1and state-2, the electronic coherence assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions in both electronic statesare assumed to have the same angular momentum (J =1,M = 0), the above TrR operatio amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two type f radi-ation, we also consid red the e↵ects of he PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the s ape of the PES of state-2. We do so by ei-ther translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Figure 3 displays the choice of the PES for thetwo electronic states, as well as the coh rence under theaction of the two types f fields.These plo s clearly show that for chaotic field xci-tations the magnitude of th co erence ⇢(t) decays ata rate much faster than that of the ultrafast coherencepulse, approaching a much lowe asy ptoti value thanthat of the tran form-limited case. This is especiallyso, as shown in the iddle row of Figu e 3, when thePES asymptotic energy separation is comparable to, orsm ller than, the spectral b ndwidth of t e two types offields. Fast decay in the incoherent excitation case alsooccurs when the Franck Co don (FC) regio s of the twosurfaces are signifi antly di↵erent. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-limited one. T us w find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figur 3) the twPES have the same asymptotic energi s, though the ini-tial decay of the oherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long times. This is understandable since at large in-ternuclear distanc s, when the electro ic states becomedegenerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to poi t out that d coherence here isnot due to environmental e↵ec b cause our molecule isisolated from n outsi environment. In the chaotic lightcase the deco rence is a result of the rando nature ofthe two wave packets and their moving away from o eanother. If, in addition, coupli g t th envir nment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tronic coherenc in mol cules is closely ti d to the co-h rent properti s f the incide t radiation. We havedone so by calculating electronic coher nces in the ph -todissociation of a molecule into a superposition of twoelectronic states, by examining two types of el ctr mag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 15 180|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Intern clear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 8 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 1|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolu -value-squared of excited nuclear wav packets as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and stat -2. Upper panel - th wave packets xcited by cha tic light, in state-1 (upper row) andstate-2 (lower row). Low r panel - the same for ex itation with a transform-limited pulse.dividing it by th to al trace of the density matrix. Inhis way we f c or out the e↵ects of the total excitatioyield that are of no i terest for our present comparativestudy.Labelling the two excited electronic states s state-1a d stat -2, the electronic coh r nce a sumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave fu ctions in both electronic statesare assumed to have the sa e angular momentum (J =1,M = 0), the b ve TrR operation amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)A ide from the compari on between two types of radi-ation, w also c nsi ered the e↵ects of the PES. Fixingth excited PES state-1 to be that of the 1u state,w vary shape of the PES of state-2. We do so by ei-ther translatin the who e PES to higher R values, or byshifting its asymptotic lectronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Fi ure 3 displays the choices of the PES for thetwo ectronic states, as well as the coherence under theaction of the two typ s of fields.These plots clearly show that for chaotic field exci-tations the magnitude of the coherence ⇢(t) d cays atrate much faster than that of the ultrafast coherencep lse, pp o ching a much lower asymptotic value thanthat of he transform-limit d case. This is esp ciallyso, as shown in the middle row of Figure 3, wh n thePES asymptotic energy separati n is compar ble to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsooccurs when the Fr nck Condon (FC) regions of the twosurfaces are significantly di↵er nt. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fractio of the coher nce generated by the ultra-fast transform-limited ne. Thus we find no evidencefor “long lived coherences” fo excitation with solar-likeh otic sources.If (as shown in the lowes panels in Figure 3) the twoPES have th same asymptotic energies, though the ini-tial decay of the coherence s much faster for excitationwith chaotic light, it approaches the same magnitudesat long imes. Th s is unde standable since at large in-ternuclear dis nc s, wh n the el ctronic states becomedegenerate, the di↵erenc in the FC regions are averagedout by our normal za ion f population in the coherencefunctio ⇢(t).It is impor ant to point out that decoherence here isot du to e vironmental e↵ects because our molecule isisolated from an o side environment. In the chaotic lightcase the decoherence is a res lt of the random nature ofthe two wave pack ts and their moving away from oneanother. If, in addition, coupling to the environment ispres nt, the electronic coherence would most likely decayat an even faster rate.C nclusions. We showed that light-induced elec-tronic coherenc in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by alculating electronic coherences in the pho-t dissociation of a molecule in o a superposition of twoelectro ic s ates, by examining two types of electromag-ne ic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 20 50 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e- 8e-183e-184e- 80 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear istance ( ohr)01 -2 -3e-184e-180 2 40 60 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01e-182e-183e-1840 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-18e-183e-184e-180 2 40 6 80 1|u(t)|2 (a.u.)I ternu lear ista ce ( ohr)01e-182e-183e- 84e-182 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)30fs90fs 150fs 210fs30fs 9 fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absol te-value-squared of x ited nuclear w v packets as a function of R at d ↵eren times, s ed 30fs apart, forexci d electro ic 1 and state-2. Upper pan l - th wave packe s excited by chaotic light, in state-1 (up er row) andstate-2 (lower row). Lower p n l - the same for excita io with a transform-limited pulse.dividing i by the total rac of the density matrix. Inthis way we factor ut he e↵ec s of th otal excitationyield that are of n interest for our present comparativestudy.Labelling th two excited electronic sta es as stat -1and state-2, the electronic coher nce assu es the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions i b t el c ronic statesare assumed to have the same angular momen um (J =1,M = 0), the above TrR ope a ion mounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comp rison b tween two types of radi-a ion, we also considered the e↵ects of the PES. Fixingthe excited PES f sta -1 to be that of th 1u stat ,we vary the shape f the PES of state-2. We d so by ei-ther ranslating the whol PES o higher R valu s, or byshifting i s asymptotic el ctronic nergy. The resultantcoherenc ⇢(t) is th n calculated according to equati n(20). Figure 3 displays th choices o the PES for thetwo electronic state , a well as coh rence und r theac ion of the wo types of fields.These plots clearly sh w that for chaot c field x i-t tions the magnitude of he coher nc ⇢(t) dec ys ata r t much faster than that of the ult afast co erencepulse, approa i g a uch l wer asymp tic value th nthat o t e tra sform-limit d ca e. This is especiallyso, as shown in the middl row f Figur 3, wh n hePES asymptotic energy separation is comparable to, orsmaller than, the spectral bandwidth of the wo types offields. Fas decay in the incoher t excit ti n case alsooccurs when the Franck ondon (FC) regio s of th twosurf c re significa ly di↵er nt. In bot ases th fi-nal el ctronic coherence indu ed by th c aotic fi ld issmall fracti n of the coherence gen rate by the ultra-fast tra sform-limit d one. Thus w find no videncefor “l ng liv d coher n es” for excitation with solar-likechaotic sources.If (as shown in the lowes pan ls in Figure 3) the twoPES have the same asymptotic energi , t ough the ini-tial de ay of the coherence is much fas er for excitationwi h chaotic light, it ppro ch s the same magnitudesa long times. This is underst ndab e since large in-ternucl ar distances, when the lectron states becomedegenerat , the di↵erence in the FC egio s ar averagedout by our normalization f populati n in the coherencefunction ⇢(t).I is im rtant to point out that decoherence h re isnot du to environmental e↵ cts b cause our molecu e isisolated from an outsid environmen . In the ch otic lightas he decoherenc is result f the rand m nature ofthe two wave packets d their moving away fr o eanother. If, in addition, coupli g t th environment isprese t, the elect onic c herence would most likely decayat an even faster rate.Conclusions. We showed that light- n uced lec-tronic coher nce in mo cules s closely ed to the co-h ent prop r es of th incide t radiation. We havedone by calcu ating electronic coher nces in the ph -todissociation of a molec le i t a superp sition of twoelectronic st tes, by exam ni g two types of electromag-netic radiation with di↵ rent degre s of coherence. We41e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 30 60 90 12 150 180|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)01 -2 -3e-184e-1820 6 80 1 0|u(t)|2 (a.u.)Internucl ar Dis ance (Boh )01e-182e-183e-184e-180 20 60 80 100|u(t)|2 (a.u.)Internucl ar Dis ance (Bohr)01e-182e-183e-184e-180 20 4 60 80 100|u(t)|2 (a.u.)Internu l ar Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-18420 4 60 80 1 0|u(t)|2 (a.u.)Internucl ar Dis ance (Boh )01e-18e-183e-184e-180 20 4 60 80 1 0|u(t)|2 (a.u.)Internu l ar Dis ance (Bohr)01e-182e-183e-184e-1820 4 60 80 100|u(t)|2 (a.u.)Internu l ar Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 3 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-173 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolut -value-squ red of excited nuclear wav packet as function of R at di↵erent times, spaced 30fs apart, forexci ed el ctronic s ate-1 n state-2. Upper panel - the ave p ckets excited by ch otic light, in state-1 (upper row) andstate-2 (low row). Lower p nel - th sa e for excitation with a transform-limited pulse.divi i g it by he otal trace of the density matrix. Ithis way we f c or out the e↵ects of he t tal xcitationyield tha are of no interest for our present comparativestudy.Labe ling h wo excited elect o ic stat s as state-1and state-2, electronic assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear w ve function in both electronic sta esare assumed o ave th same angular momen um (J =1,M = 0), the above TrR operat on amounts to int gra-tion ver R - the internucle r sep ra on,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two types of radi-a ion, w also considered the e↵ects of the PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the shape of the PES of state-2. We do so by ei-t er translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is the calculated according to equation(20). Figure 3 displays the choices of the PES for thetwo el ctronic states, as well as the coherence under theaction of the two types of fields.These plots clearly show t at for chaotic field exci-ations the magnitud of h co eren e ⇢(t) d cays ata ra much fast than that of the ul fast herencpulse, appr aching a much lower asymptotic valu thanthat f the transform-li it case. T is is p iallyso, as show in th middl row of Figure 3, h n hePES symptotic e erg separat on s c m arabl to, rsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsoccurs when the Franck Condon (FC) regions of the twosurfaces are ignificantly di↵erent. In both cases the fi-nal ele tron coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-l mited one. Thus we find no evidencefor “l g lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figure 3) the twoPES have the same asymptotic energies, though the ini-tial decay of the coherence is much faster for excitationwith chaotic ligh , it approaches the same magnitudesat ong times. This is understandable since at large in-ternuclear distances, when the electronic states becomeeg nerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to point out that decoherence here isnot due to environmental e↵ects because our molecule isisolated from an outside environment. In the chaotic lightcas the ecoherence is a result of the random nature ofthe two ve packets and their moving away from oneano r. If, in addition, coupling to the environment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tr nic coherence in molecules is closely tied to the co-herent properti s of the incident radiation. We haved n so by calculating electronic coherences in the pho-todissociat on of a m lecule into a superposition of twoelectronic states, by examining two types of electromag-n tic radiation with di↵erent degrees of coherence. We401e- 82e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184e-180 3 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1 -2 -3e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Inte nuclear Distance (Boh )01e-182e- 83e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 0 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184 10 20 40 6 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Boh )1e-18e- 83e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)3 fs90fs 150fs 210fs3 f 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: T absolute-v lu -squared of excited nuclear wave ck s as a function of R at di↵e ent times, spaced 30fs apart, forxcit d el ctronic st t -1 and state-2. U per panel - the ave packe s excited by chaotic light, in stat -1 (upper row) andstat -2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the otal tr ce of the density m trix. Inthis w y we f c or o t the e↵ ts of the total excitationyield th t are of no interest for ou present comparativestudy.Labelling he two excited electroni st t s s sta -1and state-2, the electroni assum s t e form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Si ce th nucle r wave functio s in both electronic tatesare assumed to have th sam angul r om n um (J =1,M = 0), the above TrR peratio amounts to int gra-tion over R - the inter ucle r sep ration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR1 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Asid from the comparison between two type f radi-ation, we also consid r d the e↵ects of he PES. Fixingexcited PES of state-1 to b that of the 1u state,we vary the s ape of the PES of state-2. We do so by ei-t r translating the whole PES to higher R values, or byshifting i s asymptotic electronic energy. The resultantcoh ren e ⇢(t) is then calculated according to equation(20). Figur 3 displays the choic of the PES for thetw el ctronic states, as well as the coh rence under thea ti n of the two types f fields.These plo s clearly sh w t at for chaot c field xci-tions the magnitude f t o eren e ⇢(t) cays ata rate mu h faster than that of the ultrafast coherencepulse, approaching a much l we sy ptoti value thanthat of the tran fo m-li it d case. This is especiallyso, as show in th id le r w of Figu e 3, wh n thePES asymptotic energy separation is comparable to, orsm ller than, the spectral b ndwidth of t e two types offields. F st decay in the incoherent excitation case alsoccurs when the Franck Co don (FC) regio s of the twosurfac s are signifi antly di↵erent. In both cases the fi-nal l ctronic coherence induced by the chaotic field is asmall fraction of the c herence g nerated by the ultra-fast transform-limited one. T us w find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figur 3) the twPES have the s m asymptotic e ergi s, though the ini-tial dec y of t e oherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long ti es. This is understandable since at large in-ternuclear distanc s, wh n the electro ic states becomegen t , the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to poi t out that d coherence here isnot due to enviro mental e↵ec b cause our molecule isi ola d from n outsid environment. In the chaotic lightc se the deco rence is a result of the rando nature ofthe two wave packets and their moving away from o ean ther. If, in addition, coupli g t th envir nment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclu ion . W showed that light-induced elec-ronic coher nc in mol cul s is closely ti d to the co-her n properti s f the incide t radiation. We havedone s by calculating electronic coher nces in the ph -todiss cia i n of a mol cule into a superposition of twoel ctronic stat , by examining two types of el ctr mag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 15 180|u(t)|2 (a.u.)Inte nucl ar Distance (Bohr)01 -2 -3e-184e-182 4 6 80 1|u(t)|2 (a.u.)Internuclear Distance (Boh )01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Intern clear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1842 4 6 8 1 0|u(t)|2 (a.u.)Internuclear Distance (Boh )01e-18e-183e-184e-180 20 40 60 1|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1.75e-173.5e-173 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: Th absolu -valu -squared of excited nuclear wave packets as a function of R at di↵erent times, spaced 30fs apart, forexcited l ctronic tate-1 and stat -2. Upper panel - th ave packets xcited by cha tic light, in state-1 (upper row) andst te-2 (lower row). Low r panel - the same for ex itation with a transform-limited pulse.dividing it by th o al trace of th density matrix. Inhis w y we f c or out the e↵e ts of the total excitatioyield that are of o i t rest f r our present comparativestudy.Lab ling e two ex ited elec ronic stat s s state-1a stat -2, the lectronic sumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave fu ction in both electronic sta esare assumed to ave th sa e angular momen um (J =1,M = 0), the b ve TrR operat on amounts to int gra-tion over R - the internucle r sep ration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)A ide from the compari on between two types of radi-ation, w also c nsi ered the e↵ects of the PES. Fixingth excited PES state-1 to be that of the 1u state,w vary shape of the PES of state-2. We do so by ei-ther translatin the who e PES to higher R values, or byshifting its asymptotic lectronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Fi ure 3 displays the choices of the PES for thetwo ectronic states, as well as the coherence under theaction of the two typ s of fields.These plots clearly show t at for chaotic field exci-ation the magnitu e of th co ren e ⇢(t) d cays atrate m ch fast than tha of the ultr fast coherencep lse, pp ching a mu h lower asymptotic value thanthat of h transform-li it d case. This is esp ciallyso, as how in th middle row f Figure 3, wh n hePES symp otic e e gy sepa a i n is compar ble to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsoccurs when the Fr nck Condon (FC) regions of the twosurfaces are significantly di↵er nt. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fractio of the coher nce generated by the ultra-fast transform-limited ne. Thus we find no evidencefor “long lived coherences” fo excitation with solar-likeh otic sources.If (as shown in the lowes panels in Figure 3) the twoPES have th same asymptotic energies, though the ini-tial decay of the coherence s much faster for excitationwith chaotic light, it approaches the same magnitudesat long imes. Th s is unde standable since at large in-ternuclear dis nc s, wh n the el ctronic states becomeegenerate, the di↵erenc in the FC regions are averagedout by our normal za ion f population in the coherencefunctio ⇢(t).It is impor ant to point out that decoherence here isot du to e vironmental e↵ects because our molecule isisolated from an o side environment. In the chaotic lightcase the decoherence is a res lt of the random nature ofthe two wave pack ts and their moving away from oneanother. If, in addition, coupling to the environment ispres nt, the electronic coherence would most likely decayat an even faster rate.C nclusions. We showed that light-induced elec-tronic coherenc in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by alculating electronic coherences in the pho-t dissociation of a molecule in o a superposition of twoelectro ic s ates, by examining two types of electromag-ne ic radiation with di↵erent degrees of coherence. We4013e- 84e-180 30 60 90 20 50 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e- 8e-183e-184e- 80 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear istance ( ohr)01 --3e-184e-182 40 60 80 1|u(t)|2 (a.u.)Internu lear Dista ce (Boh )01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01e- 8- 83 - 84 - 82 40 60 80 1 0|u(t)|2 (a.u.)Internu lear Dista ce (Boh )01e-18e-183e-184e- 80 2 40 6 80 1|u(t)|2 (a.u.)I ternu lear ista ce ( ohr)01e-182e-183e- 84e-182 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Inter u lear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)3 fs90fs 150fs 210fs3 f 9 fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The bsol te-value-squ red of x i ed nu lear w v packets as a f nction of R at ↵eren times, s ed 30fs apart, forexci d el ctr ic 1 a d st te-2. Uppe pan l - th ave packe s excited by chaotic light, in state-1 (up er row) andstate-2 (lower row). Lower p n l - the same for excita io with a transform-limited pulse.dividing i by the ot l rac of the d si m rix. Inthis way we f c or ut he e↵ec s of h otal excitatiyield that are of n interest for our p sent compar tivestudy.Labelling h two excit d electronic sta es a sta e-1and state-2, the electronic assu es the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the uclear ave functions i b th l c ronic statesre ssumed have th same angular momen um (J =1,M = 0), the abov TrR ope a i n mounts to i t gra-tion over R - the internuclear s paration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 | 2(R, t)|2dR. (20)Aside from the comp rison b tween two types of radi-a ion, we also con idered the e↵ects of the PES. Fixingthe xcited PES f sta -1 to be that of th 1u stat ,we vary the shape f the PES of state-2. We d so by ei-ther ranslating the whol PES o higher R valu s, or byshifting s asymptotic el ctronic nergy. The resultantcoherenc ⇢(t) is th n calculated according to equati n(20). Figure 3 displays th choices o the PES for thetwo electronic state , a well as coh rence und r theac ion of the wo types of fi lds.Thes plots clearly sh w t at for cha t c fi ld x i-tions the magnitud of coher n ⇢(t) d c ys ata r t much f s er than that of he ult afast co erencepulse, pproa i g a uch l w asymp tic value th nthat t e tra sform-li it d ca e. This is especiallyso, as show in h middl row f Fi ur 3, wh n hePES asymptotic energy s parati n is comparable to, ormaller han, the spectral bandwidth of the wo types offields. Fas decay in the incoher t excit ti n case alsoccurs when the Franck ondon (FC) regio s of th twosurf ce re significa ly di↵er nt. In bot ases th fi-nal el ctronic coherence indu ed by th c aotic fi ld issmall fracti n of the coherence gen rate by the ultra-fast ra sform-limit d one. Thus w find no videncefor “l ng liv d coher n es” for excitation with solar-likechaotic sources.If (as s wn in the lowes pan ls in Figure 3) the twoPES have the same asymptotic energi , t ough the ini-tial de ay of the coherence is much fas er for excitationwi h chaotic light, it ppro ch s the same magnitudesa long times. This is underst ndab e since large in-ternucl ar dist nces, when the lectron states becomeegenerat , the di↵ rence in the FC egio s ar averagedout by our normalization f populati n in the coherencefunction ⇢(t).I is im rtant to point out that decoherence h re isno du to environmen al e↵ cts b cause our molecu e isisolated from a o tsid environmen . In the ch otic lightas he decoherenc is result f the rand m nature ofthe two wave packe s d their moving away fr o eanother. If, n ad ition, coupli g t th environment isprese t, the elect onic c herence would most likely decayat an even faster rate.Conclusions. We showed that light- n uced lec-tronic coher nce in mo cules s closely ed to the co-h ent prop r es of th incide t radiation. We havedone by calcu ating electronic coher nces in the ph -todissociation of a molec le i t a superp sition of twoel ctr ni st t s, by exam ni g two types of electromag-netic radiation with di↵ rent degre s of coherence. WeFigure 5.3: The spatio-temporal plots of the modulus-squared radial wavefunctions of the vibrational wave packets, ψ(1,2)x (R, t). Top row is for case(A): purely repulsive excited state-1 and state-2 potentials, and bottom rowis for case (B): bound Morse wells. The functional forms for both types ofPES correspond to the first and third cases in Figure 5.5.895.1. Electronic decoherence via random vibrationwith TL pulse with incoherent pulse401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolute-valu -squared of excited nuclear wave packets as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and state-2. Upper panel - the wave packets excited by chaotic light, in state-1 (upper row) andstate-2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the total trace of the density matrix. Inthis way we factor out the e↵ects of the total excitationyield that are of no interest for our present comparativestudy.Labelling the two excited electronic states as state-1and state-2, the electronic coherence assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions in both electronic statesare assumed to have the same angular momentum (J =1,M = 0), the above TrR operation amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two types of radi-ation, we also considered the e↵ects of the PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the shape of the PES of state-2. We do so by ei-ther translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Figure 3 displays the choices of the PES for thetwo electronic states, as well as the coherence under theaction of the two types of fields.These plots clearly show that for chaotic field exci-tations the magnitude of the coherence ⇢(t) decays ata rate much faster than that of the ultrafast coherencepulse, approaching a much lower asymptotic value thanthat of the transform-limited case. This is especiallyso, as shown in the middle row of Figure 3, when thePES asymptotic energy separation is comparable to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsooccurs when the Franck Condon (FC) regions of the twosurfaces are significantly di↵erent. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-limited one. Thus we find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figure 3) the twoPES have the same asymptotic energies, though the ini-tial decay of the coherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long times. This is understandable since at large in-ternuclear distances, when the electronic states becomedegenerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to point out that decoherence here isnot due to environmental e↵ects because our molecule isisolated from an outside environment. In the chaotic lightcase the decoherence is a result of the random nature ofthe two wave packets and their moving away from oneanother. If, in addition, coupling to the environment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tronic coherence in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by calculating electronic coherences in the pho-todissociation of a molecule into a superposition of twoelectronic states, by examining two types of electromag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolute-value-squared of excited nuclear wave ck ts as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and state-2. Upper panel - the wave packe s excited by chaotic light, in stat -1 (upper row) andstate-2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the total trace of the density matrix. Inthis way we factor out the e↵ects of the total excitationyield that are of no interest for our present comparativestudy.Labelling the two excited electronic states as state-1and state-2, the electronic coherence assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions in both electronic statesare assumed to have the same angular momentum (J =1,M = 0), the above TrR operatio amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two type f radi-ation, we also consid red the e↵ects of he PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the s ape of the PES of state-2. We do so by ei-ther translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Figure 3 displays the choice of the PES for thetwo electronic states, as well as the coh rence under theaction of the two types f fields.These plo s clearly show that for chaotic field xci-tations the magnitude of th co erence ⇢(t) decays ata rate much faster than that of the ultrafast coherencepulse, approaching a much lowe asy ptoti value thanthat of the tran form-limited case. This is especiallyso, as shown in the iddle row of Figu e 3, when thePES asymptotic energy separation is comparable to, orsm ller than, the spectral b ndwidth of t e two types offields. Fast decay in the incoherent excitation case alsooccurs when the Franck Co don (FC) regio s of the twosurfaces are signifi antly di↵erent. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-limited one. T us w find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figur 3) the twPES have the same asymptotic energi s, though the ini-tial decay of the oherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long times. This is understandable since at large in-ternuclear distanc s, when the electro ic states becomedegenerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to poi t out that d coherence here isnot due to environmental e↵ec b cause our molecule isisolated from n outsi environment. In the chaotic lightcase the deco rence is a result of the rando nature ofthe two wave packets and their moving away from o eanother. If, in addition, coupli g t th envir nment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tronic coherenc in mol cules is closely ti d to the co-h rent properti s f the incide t radiation. We havedone so by calculating electronic coher nces in the ph -todissociation of a molecule into a superposition of twoelectronic states, by examining two types of el ctr mag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 15 180|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01 -2 -3e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Intern clear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1840 20 40 60 8 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 20 40 60 1|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolu -value-squared of excited nuclear wav packets as a function of R at di↵erent times, spaced 30fs apart, forexcited electronic state-1 and stat -2. Upper panel - th wave packets xcited by cha tic light, in state-1 (upper row) andstate-2 (lower row). Low r panel - the same for ex itation with a transform-limited pulse.dividing it by th to al trace of the density matrix. Inhis way we f c or out the e↵ects of the total excitatioyield that are of no i terest for our present comparativestudy.Labelling the two excited electronic states s state-1a d stat -2, the electronic coh r nce a sumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave fu ctions in both electronic statesare assumed to have the sa e angular momentum (J =1,M = 0), the b ve TrR operation amounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)A ide from the compari on between two types of radi-ation, w also c nsi ered the e↵ects of the PES. Fixingth excited PES state-1 to be that of the 1u state,w vary shape of the PES of state-2. We do so by ei-ther translatin the who e PES to higher R values, or byshifting its asymptotic lectronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Fi ure 3 displays the choices of the PES for thetwo ectronic states, as well as the coherence under theaction of the two typ s of fields.These plots clearly show that for chaotic field exci-tations the magnitude of the coherence ⇢(t) d cays atrate much faster than that of the ultrafast coherencep lse, pp o ching a much lower asymptotic value thanthat of he transform-limit d case. This is esp ciallyso, as shown in the middle row of Figure 3, wh n thePES asymptotic energy separati n is compar ble to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsooccurs when the Fr nck Condon (FC) regions of the twosurfaces are significantly di↵er nt. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fractio of the coher nce generated by the ultra-fast transform-limited ne. Thus we find no evidencefor “long lived coherences” fo excitation with solar-likeh otic sources.If (as shown in the lowes panels in Figure 3) the twoPES have th same asymptotic energies, though the ini-tial decay of the coherence s much faster for excitationwith chaotic light, it approaches the same magnitudesat long imes. Th s is unde standable since at large in-ternuclear dis nc s, wh n the el ctronic states becomedegenerate, the di↵erenc in the FC regions are averagedout by our normal za ion f population in the coherencefunctio ⇢(t).It is impor ant to point out that decoherence here isot du to e vironmental e↵ects because our molecule isisolated from an o side environment. In the chaotic lightcase the decoherence is a res lt of the random nature ofthe two wave pack ts and their moving away from oneanother. If, in addition, coupling to the environment ispres nt, the electronic coherence would most likely decayat an even faster rate.C nclusions. We showed that light-induced elec-tronic coherenc in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by alculating electronic coherences in the pho-t dissociation of a molecule in o a superposition of twoelectro ic s ates, by examining two types of electromag-ne ic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 20 50 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e- 8e-183e-184e- 80 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear istance ( ohr)01 -2 -3e-184e-180 2 40 60 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01e-182e-183e-1840 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-18e-183e-184e-180 2 40 6 80 1|u(t)|2 (a.u.)I ternu lear ista ce ( ohr)01e-182e-183e- 84e-182 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)30fs90fs 150fs 210fs30fs 9 fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absol te-value-squared of x ited nuclear w v packets as a function of R at d ↵eren times, s ed 30fs apart, forexci d electro ic 1 and state-2. Upper pan l - th wave packe s excited by chaotic light, in state-1 (up er row) andstate-2 (lower row). Lower p n l - the same for excita io with a transform-limited pulse.dividing i by the total rac of the density matrix. Inthis way we factor ut he e↵ec s of th otal excitationyield that are of n interest for our present comparativestudy.Labelling th two excited electronic sta es as stat -1and state-2, the electronic coher nce assu es the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave functions i b t el c ronic statesare assumed to have the same angular momen um (J =1,M = 0), the above TrR ope a ion mounts to integra-tion over R - the internuclear separation,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comp rison b tween two types of radi-a ion, we also considered the e↵ects of the PES. Fixingthe excited PES f sta -1 to be that of th 1u stat ,we vary the shape f the PES of state-2. We d so by ei-ther ranslating the whol PES o higher R valu s, or byshifting i s asymptotic el ctronic nergy. The resultantcoherenc ⇢(t) is th n calculated according to equati n(20). Figure 3 displays th choices o the PES for thetwo electronic state , a well as coh rence und r theac ion of the wo types of fields.These plots clearly sh w that for chaot c field x i-t tions the magnitude of he coher nc ⇢(t) dec ys ata r t much faster than that of the ult afast co erencepulse, approa i g a uch l wer asymp tic value th nthat o t e tra sform-limit d ca e. This is especiallyso, as shown in the middl row f Figur 3, wh n hePES asymptotic energy separation is comparable to, orsmaller than, the spectral bandwidth of the wo types offields. Fas decay in the incoher t excit ti n case alsooccurs when the Franck ondon (FC) regio s of th twosurf c re significa ly di↵er nt. In bot ases th fi-nal el ctronic coherence indu ed by th c aotic fi ld issmall fracti n of the coherence gen rate by the ultra-fast tra sform-limit d one. Thus w find no videncefor “l ng liv d coher n es” for excitation with solar-likechaotic sources.If (as shown in the lowes pan ls in Figure 3) the twoPES have the same asymptotic energi , t ough the ini-tial de ay of the coherence is much fas er for excitationwi h chaotic light, it ppro ch s the same magnitudesa long times. This is underst ndab e since large in-ternucl ar distances, when the lectron states becomedegenerat , the di↵erence in the FC egio s ar averagedout by our normalization f populati n in the coherencefunction ⇢(t).I is im rtant to point out that decoherence h re isnot du to environmental e↵ cts b cause our molecu e isisolated from an outsid environmen . In the ch otic lightas he decoherenc is result f the rand m nature ofthe two wave packets d their moving away fr o eanother. If, in addition, coupli g t th environment isprese t, the elect onic c herence would most likely decayat an even faster rate.Conclusions. We showed that light- n uced lec-tronic coher nce in mo cules s closely ed to the co-h ent prop r es of th incide t radiation. We havedone by calcu ating electronic coher nces in the ph -todissociation of a molec le i t a superp sition of twoelectronic st tes, by exam ni g two types of electromag-netic radiation with di↵ rent degre s of coherence. We41e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-18e-183e-184e-180 30 60 90 12 150 180|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)01 -2 -3e-184e-1820 6 80 1 0|u(t)|2 (a.u.)Internucl ar Dis ance (Boh )01e-182e-183e-184e-180 20 60 80 100|u(t)|2 (a.u.)Internucl ar Dis ance (Bohr)01e-182e-183e-184e-180 20 4 60 80 100|u(t)|2 (a.u.)Internu l ar Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-18420 4 60 80 1 0|u(t)|2 (a.u.)Internucl ar Dis ance (Boh )01e-18e-183e-184e-180 20 4 60 80 1 0|u(t)|2 (a.u.)Internu l ar Dis ance (Bohr)01e-182e-183e-184e-1820 4 60 80 100|u(t)|2 (a.u.)Internu l ar Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 3 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-173 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The absolut -value-squ red of excited nuclear wav packet as function of R at di↵erent times, spaced 30fs apart, forexci ed el ctronic s ate-1 n state-2. Upper panel - the ave p ckets excited by ch otic light, in state-1 (upper row) andstate-2 (low row). Lower p nel - th sa e for excitation with a transform-limited pulse.divi i g it by he otal trace of the density matrix. Ithis way we f c or out the e↵ects of he t tal xcitationyield tha are of no interest for our present comparativestudy.Labe ling h wo excited elect o ic stat s as state-1and state-2, electronic assumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear w ve function in both electronic sta esare assumed o ave th same angular momen um (J =1,M = 0), the above TrR operat on amounts to int gra-tion ver R - the internucle r sep ra on,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Aside from the comparison between two types of radi-a ion, w also considered the e↵ects of the PES. Fixingthe excited PES of state-1 to be that of the 1u state,we vary the shape of the PES of state-2. We do so by ei-t er translating the whole PES to higher R values, or byshifting its asymptotic electronic energy. The resultantcoherence ⇢(t) is the calculated according to equation(20). Figure 3 displays the choices of the PES for thetwo el ctronic states, as well as the coherence under theaction of the two types of fields.These plots clearly show t at for chaotic field exci-ations the magnitud of h co eren e ⇢(t) d cays ata ra much fast than that of the ul fast herencpulse, appr aching a much lower asymptotic valu thanthat f the transform-li it case. T is is p iallyso, as show in th middl row of Figure 3, h n hePES symptotic e erg separat on s c m arabl to, rsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsoccurs when the Franck Condon (FC) regions of the twosurfaces are ignificantly di↵erent. In both cases the fi-nal ele tron coherence induced by the chaotic field is asmall fraction of the coherence generated by the ultra-fast transform-l mited one. Thus we find no evidencefor “l g lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figure 3) the twoPES have the same asymptotic energies, though the ini-tial decay of the coherence is much faster for excitationwith chaotic ligh , it approaches the same magnitudesat ong times. This is understandable since at large in-ternuclear distances, when the electronic states becomeeg nerate, the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to point out that decoherence here isnot due to environmental e↵ects because our molecule isisolated from an outside environment. In the chaotic lightcas the ecoherence is a result of the random nature ofthe two ve packets and their moving away from oneano r. If, in addition, coupling to the environment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclusions. We showed that light-induced elec-tr nic coherence in molecules is closely tied to the co-herent properti s of the incident radiation. We haved n so by calculating electronic coherences in the pho-todissociat on of a m lecule into a superposition of twoelectronic states, by examining two types of electromag-n tic radiation with di↵erent degrees of coherence. We401e- 82e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184e-180 3 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)1 -2 -3e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Inte nuclear Distance (Boh )01e-182e- 83e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 0 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1e-182e- 83e-184 10 20 40 6 80 1 0|u(t)|2 (a.u.)Internuclear Distance (Boh )1e-18e- 83e-184e-180 20 40 60 80 1 0|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-1820 40 60 80 100|u(t)|2 (a.u.)Inte nuclear Distance (Bohr)1e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)3 fs90fs 150fs 210fs3 f 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: T absolute-v lu -squared of excited nuclear wave ck s as a function of R at di↵e ent times, spaced 30fs apart, forxcit d el ctronic st t -1 and state-2. U per panel - the ave packe s excited by chaotic light, in stat -1 (upper row) andstat -2 (lower row). Lower panel - the same for excitation with a transform-limited pulse.dividing it by the otal tr ce of the density m trix. Inthis w y we f c or o t the e↵ ts of the total excitationyield th t are of no interest for ou present comparativestudy.Labelling he two excited electroni st t s s sta -1and state-2, the electroni assum s t e form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Si ce th nucle r wave functio s in both electronic tatesare assumed to have th sam angul r om n um (J =1,M = 0), the above TrR peratio amounts to int gra-tion over R - the inter ucle r sep ration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR1 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)Asid from the comparison between two type f radi-ation, we also consid r d the e↵ects of he PES. Fixingexcited PES of state-1 to b that of the 1u state,we vary the s ape of the PES of state-2. We do so by ei-t r translating the whole PES to higher R values, or byshifting i s asymptotic electronic energy. The resultantcoh ren e ⇢(t) is then calculated according to equation(20). Figur 3 displays the choic of the PES for thetw el ctronic states, as well as the coh rence under thea ti n of the two types f fields.These plo s clearly sh w t at for chaot c field xci-tions the magnitude f t o eren e ⇢(t) cays ata rate mu h faster than that of the ultrafast coherencepulse, approaching a much l we sy ptoti value thanthat of the tran fo m-li it d case. This is especiallyso, as show in th id le r w of Figu e 3, wh n thePES asymptotic energy separation is comparable to, orsm ller than, the spectral b ndwidth of t e two types offields. F st decay in the incoherent excitation case alsoccurs when the Franck Co don (FC) regio s of the twosurfac s are signifi antly di↵erent. In both cases the fi-nal l ctronic coherence induced by the chaotic field is asmall fraction of the c herence g nerated by the ultra-fast transform-limited one. T us w find no evidencefor “long lived coherences” for excitation with solar-likechaotic sources.If (as shown in the lowest panels in Figur 3) the twPES have the s m asymptotic e ergi s, though the ini-tial dec y of t e oherence is much faster for excitationwith chaotic light, it approaches the same magnitudesat long ti es. This is understandable since at large in-ternuclear distanc s, wh n the electro ic states becomegen t , the di↵erence in the FC regions are averagedout by our normalization of population in the coherencefunction ⇢(t).It is important to poi t out that d coherence here isnot due to enviro mental e↵ec b cause our molecule isi ola d from n outsid environment. In the chaotic lightc se the deco rence is a result of the rando nature ofthe two wave packets and their moving away from o ean ther. If, in addition, coupli g t th envir nment ispresent, the electronic coherence would most likely decayat an even faster rate.Conclu ion . W showed that light-induced elec-ronic coher nc in mol cul s is closely ti d to the co-her n properti s f the incide t radiation. We havedone s by calculating electronic coher nces in the ph -todiss cia i n of a mol cule into a superposition of twoel ctronic stat , by examining two types of el ctr mag-netic radiation with di↵erent degrees of coherence. We401e-182e-183e-184e-180 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 30 60 90 120 15 180|u(t)|2 (a.u.)Inte nucl ar Distance (Bohr)01 -2 -3e-184e-182 4 6 80 1|u(t)|2 (a.u.)Internuclear Distance (Boh )01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Intern clear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-1842 4 6 8 1 0|u(t)|2 (a.u.)Internuclear Distance (Boh )01e-18e-183e-184e-180 20 40 60 1|u(t)|2 (a.u.)Internucl ar Distance (Bohr)01e-182e-183e-184e-1820 40 60 80 10|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e-182e-183e-184e-180 20 40 60 80 100|u(t)|2 (a.u.)Internuclear Distance (Bohr)1.75e-173.5e-173 60 90 120 150 180|u(t)|2 (a.u.)Inter uclear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)30fs90fs 150fs 210fs30fs 90fs 150fs 210fs30fs300fs30fs300fsFIG. 2: Th absolu -valu -squared of excited nuclear wave packets as a function of R at di↵erent times, spaced 30fs apart, forexcited l ctronic tate-1 and stat -2. Upper panel - th ave packets xcited by cha tic light, in state-1 (upper row) andst te-2 (lower row). Low r panel - the same for ex itation with a transform-limited pulse.dividing it by th o al trace of th density matrix. Inhis w y we f c or out the e↵e ts of the total excitatioyield that are of o i t rest f r our present comparativestudy.Lab ling e two ex ited elec ronic stat s s state-1a stat -2, the lectronic sumes the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the nuclear wave fu ction in both electronic sta esare assumed to ave th sa e angular momen um (J =1,M = 0), the b ve TrR operat on amounts to int gra-tion over R - the internucle r sep ration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 |u2(R, t)|2dR. (20)A ide from the compari on between two types of radi-ation, w also c nsi ered the e↵ects of the PES. Fixingth excited PES state-1 to be that of the 1u state,w vary shape of the PES of state-2. We do so by ei-ther translatin the who e PES to higher R values, or byshifting its asymptotic lectronic energy. The resultantcoherence ⇢(t) is then calculated according to equation(20). Fi ure 3 displays the choices of the PES for thetwo ectronic states, as well as the coherence under theaction of the two typ s of fields.These plots clearly show t at for chaotic field exci-ation the magnitu e of th co ren e ⇢(t) d cays atrate m ch fast than tha of the ultr fast coherencep lse, pp ching a mu h lower asymptotic value thanthat of h transform-li it d case. This is esp ciallyso, as how in th middle row f Figure 3, wh n hePES symp otic e e gy sepa a i n is compar ble to, orsmaller than, the spectral bandwidth of the two types offields. Fast decay in the incoherent excitation case alsoccurs when the Fr nck Condon (FC) regions of the twosurfaces are significantly di↵er nt. In both cases the fi-nal electronic coherence induced by the chaotic field is asmall fractio of the coher nce generated by the ultra-fast transform-limited ne. Thus we find no evidencefor “long lived coherences” fo excitation with solar-likeh otic sources.If (as shown in the lowes panels in Figure 3) the twoPES have th same asymptotic energies, though the ini-tial decay of the coherence s much faster for excitationwith chaotic light, it approaches the same magnitudesat long imes. Th s is unde standable since at large in-ternuclear dis nc s, wh n the el ctronic states becomeegenerate, the di↵erenc in the FC regions are averagedout by our normal za ion f population in the coherencefunctio ⇢(t).It is impor ant to point out that decoherence here isot du to e vironmental e↵ects because our molecule isisolated from an o side environment. In the chaotic lightcase the decoherence is a res lt of the random nature ofthe two wave pack ts and their moving away from oneanother. If, in addition, coupling to the environment ispres nt, the electronic coherence would most likely decayat an even faster rate.C nclusions. We showed that light-induced elec-tronic coherenc in molecules is closely tied to the co-herent properties of the incident radiation. We havedone so by alculating electronic coherences in the pho-t dissociation of a molecule in o a superposition of twoelectro ic s ates, by examining two types of electromag-ne ic radiation with di↵erent degrees of coherence. We4013e- 84e-180 30 60 90 20 50 180|u(t)|2 (a.u.)Internuclear Distance (Bohr)01e- 8e-183e-184e- 80 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear istance ( ohr)01 --3e-184e-182 40 60 80 1|u(t)|2 (a.u.)Internu lear Dista ce (Boh )01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01e- 8- 83 - 84 - 82 40 60 80 1 0|u(t)|2 (a.u.)Internu lear Dista ce (Boh )01e-18e-183e-184e- 80 2 40 6 80 1|u(t)|2 (a.u.)I ternu lear ista ce ( ohr)01e-182e-183e- 84e-182 40 6 80 10|u(t)|2 (a.u.)Internu lear Dista ce (Bohr)01e-182e-183e-184e-180 2 40 60 80 100|u(t)|2 (a.u.)Internu lear Distance (Bohr)01.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Inter u lear Distance (Bohr)1.75e-173.5e-170 30 60 90 120 150 180|u(t)|2 (a.u.)Internu lear Distance (Bohr)3 fs90fs 150fs 210fs3 f 9 fs 150fs 210fs30fs300fs30fs300fsFIG. 2: The bsol te-value-squ red of x i ed nu lear w v packets as a f nction of R at ↵eren times, s ed 30fs apart, forexci d el ctr ic 1 a d st te-2. Uppe pan l - th ave packe s excited by chaotic light, in state-1 (up er row) andstate-2 (lower row). Lower p n l - the same for excita io with a transform-limited pulse.dividing i by the ot l rac of the d si m rix. Inthis way we f c or ut he e↵ec s of h otal excitatiyield that are of n interest for our p sent compar tivestudy.Labelling h two excit d electronic sta es a sta e-1and state-2, the electronic assu es the form,⇢(t) ⌘TrR[⇢21(R, t)]TrR[⇢11(R, t)] + TrR[⇢22(R, t)].Since the uclear ave functions i b th l c ronic statesre ssumed have th same angular momen um (J =1,M = 0), the abov TrR ope a i n mounts to i t gra-tion over R - the internuclear s paration,⇢(t) =R11 u⇤1(R, t)u2(R, t)dRR11 |u1(R, t)|2dR +R11 | 2(R, t)|2dR. (20)Aside from the comp rison b tween two types of radi-a ion, we also con idered the e↵ects of the PES. Fixingthe xcited PES f sta -1 to be that of th 1u stat ,we vary the shape f the PES of state-2. We d so by ei-ther ranslating the whol PES o higher R valu s, or byshifting s asymptotic el ctronic nergy. The resultantcoherenc ⇢(t) is th n calculated according to equati n(20). Figure 3 displays th choices o the PES for thetwo electronic state , a well as coh rence und r theac ion of the wo types of fi lds.Thes plots clearly sh w t at for cha t c fi ld x i-tions the magnitud of coher n ⇢(t) d c ys ata r t much f s er than that of he ult afast co erencepulse, pproa i g a uch l w asymp tic value th nthat t e tra sform-li it d ca e. This is especiallyso, as show in h middl row f Fi ur 3, wh n hePES asymptotic energy s parati n is comparable to, ormaller han, the spectral bandwidth of the wo types offields. Fas decay in the incoher t excit ti n case alsoccurs when the Franck ondon (FC) regio s of th twosurf ce re significa ly di↵er nt. In bot ases th fi-nal el ctronic coherence indu ed by th c aotic fi ld issmall fracti n of the coherence gen rate by the ultra-fast ra sform-limit d one. Thus w find no videncefor “l ng liv d coher n es” for excitation with solar-likechaotic sources.If (as s wn in the lowes pan ls in Figure 3) the twoPES have the same asymptotic energi , t ough the ini-tial de ay of the coherence is much fas er for excitationwi h chaotic light, it ppro ch s the same magnitudesa long times. This is underst ndab e since large in-ternucl ar dist nces, when the lectron states becomeegenerat , the di↵ rence in the FC egio s ar averagedout by our normalization f populati n in the coherencefunction ⇢(t).I is im rtant to point out that decoherence h re isno du to environmen al e↵ cts b cause our molecu e isisolated from a o tsid environmen . In the ch otic lightas he decoherenc is result f the rand m nature ofthe two wave packe s d their moving away fr o eanother. If, n ad ition, coupli g t th environment isprese t, the elect onic c herence would most likely decayat an even faster rate.Conclusions. We showed that light- n uced lec-tronic coher nce in mo cules s closely ed to the co-h ent prop r es of th incide t radiation. We havedone by calcu ating electronic coher nces in the ph -todissociation of a molec le i t a superp sition of twoel ctr ni st t s, by exam ni g two types of electromag-netic radiation with di↵ rent degre s of coherence. WeFigure 5.4: Same as Figure 5.3, but for photoexcitation with a single real-ization of the incoherent pulse.905.1. Electronic decoherence via random vibrationconvenient to normalize ρ1,2(t) by dividing it by ρ1,1(t) +ρ2,2(t), the densitymatrix trace, givingC(t) = ρ1,2(t)ρ1,1(t) + ρ2,2(t). (5.21)In this way we factor out the effects of the total excitation yield that areof no interest for this comparative study. Note also that equation (5.20),by averaging over vibrations, includes the effect of decoherence due to thevibrational degree of freedom on the electronic coherences. The resultantnormalized coherences C(t) are plotted in Figure 5.5, for different choicesof the PES for the two electronic states, under the action of the two typesof fields. Interestingly, in the case of incoherent light, convergence for theresults shown were obtained with averaging over only ten realizations of theelectric field.These plots clearly show that for incoherent field excitations the mag-nitude of the coherence C(t) decays at a much faster rate than that of thecoherent pulse, approaching a much lower asymptotic value than that of thecoherent case. This is especially so, as shown in the middle row of Figure5.5, when coherence C(t) is not significant to begin with, in the coherentcase, due to small Franck-Condon overlap regions of the two surfaces. In thecase of the bound Morse potential wells, clear and gradual decoherence ofthe two wave packets is observed for the case of coherent excitation. This isin strong contrast with the incoherent pulse excitation, where the coherenceC(t) decays almost immediately.In all of the three cases, the final electronic coherence induced by theincoherent field is a small fraction of the coherence generated by the co-herent case. Thus we find that the incoherence of the light eliminates longlived coherences when the excitation is carried out with pulses of solar-likeincoherent sources.Several comments are in order. First, note that decoherence observedhere does not include additional environmental effects, since the molecule isisolated, i.e. the system is closed. Rather, the observed fast coherence lossarises, in the incoherent light case, from the nature of the two wave packetsas they lose mutual coherence. That is, the molecular vibration serves as the915.1. Electronic decoherence via random vibration-0.100.000.100.200.30 0  1  2  3  4  5  6  7  8  9  10Energy (Hartree)Internuclear Distance (Bohr)ground statestate-1state-2-0.100.000.100.200.30 0  1  2  3  4  5  6  7  8  9  10Energy (Hartree)Internuclear Distance (Bohr)ground statestate-1state-2-0.100.000.100.200.30 0  5  10  15  20Energy (Hartree)Internuclear Distance (Bohr)ground statestate-1state-20.000.100.200.300.400.50 0  50  100  150  200  250|C(t)|Time (fs)with incoherent lightwith transform-limited light0.000.100.200.300.400.50 0  50  100  150  200  250|C(t)|Time (fs)with incoherent lightwith transform-limited light0.000.100.200.300.400.50 0  50  100  150  200  250|C(t)|Time (fs)with incoherent lightwith transform-limited lightFigure 5.5: The left column shows three pairs of PES’s, and the v = 5vibrational wave function. The repulsive PES on the top has functionalform W (i)x (R) = W (i)0 exp[−(R − R(i)0 )/a(i)] + W(i)∞ , and the Morse poten-tials in the bottom panel have W (i)x (R) = W (i)0[1− e−(R−R(i)0 )/a(i)]2 +W (i)∞ ,both described by a parameter vector vn = (W (n)0 , R(n)0 , a(n),W(n)∞ ). For thetop panel, v1 = (0.1, 3.3, 1.0, 0), v2 = (0.1, 3.0, 1.2, 0.005). In the middlepanel, v1 is the same but v2 = (0.1, 2.5, 1.0, 0.01). In the bottom panel,v1 = (0.1, 5.0, 2.5, 0) and v2 = (0.1, 5.01, 2.53, 0.01). All numerical valuesare in atomic units. In the right column is the absolute value of C(t), thenormalized electronic coherence resulting from coherent and incoherent lightexcitation for the three different PES of the left column. The results for theincoherent light were averaged over a set of 10 realizations. (Additionalrealizations did not significantly alter the results.)925.1. Electronic decoherence via random vibrationdecohering environment [108, 109], and it is particularly effective in this casedue to the highly unstructured and choppy nuclear wave functions createdby incoherent light. If, then, we consider low degree of coherence as intrue solar radiation, or additional coupling to an external environment, thedecay of electronic coherence will reflect the change accordingly [100], andas a consequence, be expected to be much faster, and to levels much closerto zero. Long lived electronic coherences are thus not expected to persist inrealistic open systems irradiated with incoherent sources.Second, note that the incoherent light source used above acts over a 200fs time scale. Thus, although the short coherence time of natural incoherentlight is represented in this computation, there are two significant differencesbetween the results here and that which would result from true solar radi-ation, which acts over far longer times (effectively CW): (1) Due to, again,the smooth pulse envelope, the incoherent pulse used here possesses morecoherence than would natural solar radiation that is incident for minutes orlonger [95]; (2) At such long times, as discussed elsewhere [95, 98], long timeexcitation of isolated molecules using natural incoherent light leads, whenaveraged over realizations, to stationary eigenstates of the Hamiltonian thatdo not evolve in time. Relaxation in open systems also leads to mixtures ofstationary states [100].5.1.4 ConclusionsThis study has shown that the nature of the light-induced wave function ofmolecular vibration, as well as the resultant electronic (de)coherence in themolecule, is strongly tied to the coherence properties of the incident radi-ation. Coherent light pulses produce well localized wave packets, whereasincoherent pulses produce irregularly structured coordinate space densities.In addition, the electronic coherence is quantified in the photoexcitation ofa molecule into a superposition of two uncoupled electronic states, using thetwo types of light pulses with different degrees of coherence. The resultsshow decreased electronic coherence values and much faster decoherencerates when the excitation is conducted with pulsed incoherent light than935.2. Vibrational resonances excited by incoherent lightwith coherent pulses of the same central frequency and spectral bandwidth.We would also expect that additional coupling between the excited electronicstates, which makes the wave packet dynamics non-adiabatic, could not im-prove the electronic coherence. This is because we can either switch to theadiabatic potentials and arrive at similar results, or consider that the phasesof the wave packet in each diabatic channel would be only randomized evenmore through the coupling between the electronic manifolds.Thus, even in the absence of external environmental effects, the randomcharacter of the incoherent light is imparted directly into the molecular vi-bration, and enhances electronic decoherence. This study adds considerablesupport to the conclusion [98, 100, 110] that excitation with coherent vsincoherent light results in substantially different dynamical behaviour. Assuch, modern coherent pulsed laser experiments on biological molecules dogive insight into the nature of the system and its coupling to the surroundingenvironment, but do not necessarily imply that similar dynamical behaviourwill be observed under natural illumination with incoherent light of the samespectral profiles.5.2 Vibrational resonances excited by incoherentlight5.2.1 MotivationAs the previous study has shown, the quantum states created by pulsedcoherent vs. incoherent photoexcitation are drastically different, in termsof the spatio-temporal dynamics of the wave functions. The resultant elec-tronic coherence determined by the vibration would then be destroyed ona much faster pace and to a much lower level. This study continues onthe theme of contrasting coherent vs. incoherent molecular vibration, andstresses some important aspects not yet covered: (1) the statistically aver-aged spatio-temporal dynamics of vibration, and (2) vibrations of coupledbound-continuum nature, and the accompanying dynamics of energy eigen-states following coherent and incoherent photoexcitation. This study then945.2. Vibrational resonances excited by incoherent lightserves both as a detailed extension and support for the previous study, wherealso a different type of optical incoherent light is used, and as a steppingstone to future research on more elaborate analysis of quantum state dy-namics under realistic solar radiation.5.2.2 Excited-state dynamics of IBrIn the previous study, we looked at various model potential energy sur-faces (PES), for the energy-degenerate electronic manifolds, into which themolecule is excited by coherent and incoherent light. These pairs of PES’s,without any electronic coupling, support vibrations of either bound or con-tinuum nature. However, more interesting and not previously discussed is acombination of coupled bound and continuum PES’s. The bound manifoldsupports localized coherent quantum wave packet oscillation (the “closedchannel”), whereas the continuum manifold allows only dissociation (the“open channel”). Due to the coupling, population in the closed channel canspontaneously flow to the continuum, and the vibrational eigenstate to thebound manifold effectively acquires a finite lifetime due to the interactionwith the open channel. The mixed manifold, viewed as a whole, can thenhost total vibrational eigenstates as superpositions of bound and continuumstates from each individual manifold. Consequently, the optical transitioninto such total vibrational eigenstates can exhibit sharp peaks as a functionof energy, and the original bound eigenstates become so-called vibrationalresonances.The iodine bromide (IBr) molecule has a pair of well-understood elec-tronic excited states that serve exactly as the ideal setup for such photoexci-tation dynamics of vibrational resonances. The two excited electronic man-ifolds are known in the diabatic representation as the Y (O+) and B(3ΠO+)states, which we will simply refer to as Y and B. At appropriate exci-tation energies, the B state is bounded and supports localized vibrationalstates, while the Y state is purely repulsive and hosts only continuum states.Their potential energy surfaces in the iodine-bromine bond length degree offreedom, as shown in Figure 5.6, overlap in energy, and interact with inter-955.2. Vibrational resonances excited by incoherent light-0.1-0.0500.050.122 22.5 23 23.5 24 24.5 25Energy (Hartree)R (Bohr)-0.1-0.0500.050.14 5 6 7 8 9 10Energy (Hartree)R (Bohr)-0.1-0.0500.050.14 5 6 7 8 9 10Energy (Hartree)R (Bohr)YB-0.1-0.0500.050.14 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25Energy (Hartree)R (Bohr)-0.1-0.0500.050.14 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2 23 24 25Energy (Hartree)R (Bohr)-0.1-0.0500.050.14 5 6 7 8 9 10Energy (Hartree)R (Bohr)YBTuesday, 30 July, 13Figure 5.6: The potential energy surfaces for the ground, Y and B states ofIBr. Shown in the R = 23− 25 Bohr range are the magnitudes of negativeimaginary potentials, artificially added as numerical absorbing boundaries,of functional form Vabsorb(R) = −i(0.075a.u.)(R− 23Bohr)2.mediate strength [111], enabling the molecule pre-excited in the bounded Bstate to transition into the open Y state under time evolution, upon whichthe bond length diverges, and the molecule dissociates.For the dynamics analysis, we represent the total molecular quantumstate by three time-dependent vibrational wave packets in the basis of thethree relevant electronic states. The molecule resides completely in theground vibrational eigenstate φ0(R) of the J = 0 ground electronic statebefore the onset of any light field (t = 0),Ψ(t, R) = 1Rψg(t, R)ψY (t, R)ψB(t, R) , Ψ(t = 0, R) = 1Rφ0(R)00 . (5.22)The photoexcitation is assumed to be a first-order perturbation, and anyhigher-order electronic or rotational (J > 1) excitations are ignored. Theoptical energies needed to reach resonances in the Y/B mixed manifoldfrom φ0(R), the approximate (R-independent) coupling between the twomanifolds, as well as the resonance line-shapes, are taken from Ref. [111].The total quantum state of the molecule then evolves in time according965.2. Vibrational resonances excited by incoherent lightto the Schro¨dinger’s equation,i~ ddtΨ(t, R) = Hˆ(t, R)Ψ(t, R), (5.23)where the full Hamiltonian matrix is (although it can be simplified in theperturbative limit [112])Hˆ =TˆN +Wg(R) −VY,g(R)E(t) −VB,g(R)E(t)−V ∗Y,g(R)E(t) TˆN +WY (R) VB,Y (R)−V ∗B,g(R)E(t) V ∗B,Y (R) TˆN +WB(R) , (5.24)with the usual kinetic energy operator in 3D (m being the reduced mass ofiodine and bromine atom)TˆN = −~22m( d2dR2 +2RddR).This then sets up the necessary formulation for the wave packet dynamics,where the electric field E(t) can again be random.5.2.3 Spectral incoherenceThere are many physical mechanisms which produce incoherent light. In theprevious study, we modelled the field, in the time domain, as if it comes froma single atomic emitter, which has a finite total duration, rather flat intensityprofile (see Figure 5.1), but suffers sudden jumps in phase and frequency,with random amount, at random time intervals. The largest magnitude ofthe jumps and the total duration determine the field’s autocorrelation, andhence its spectral properties [103]. Here, on the other hand, we model theincoherent light in the spectral domain. Such field can be thought of ascoming from a purely coherent source, but having spectral phases random-ized at each frequency in its spectrum. Assuming the source has a uniquelinear polarization (e.g. having passed through a polarizer), we can write975.2. Vibrational resonances excited by incoherent lightthe scalar electric field, following above description, asE0(t) =∫ ∞−∞s0(ω)eiΦ(ω)e−iωtdω (5.25)where s0(ω) is a real, deterministic function, but the phase angles Φ(ω) arerandom variables at each frequency ω, uniformly distributed across [−pi, pi],with delta-function cross-correlation 〈ei[Φ(ω)−Φ(ω′)]〉 = δ(ω − ω′).Due to the randomness in its spectral phases, this incoherent field canin fact stretch indefinitely in time. Its power spectrum can then be calcu-lated using the Wiener-Khinchin Theorem [103], as the Fourier transformF [Γ(τ)](ω) of the field autocorrelation functionΓ(τ) = 〈E∗0(t)E0(t+ τ)〉=∫ ∞−∞∫ ∞−∞|s0(ω′)||s0(ω)|〈ei[Φ(ω′)−Φ(ω)]〉ei(ω′−ω)te−iωτdω′dω=∫ ∞−∞|s0(ω)|2e−iωτdω = F−1[|s0(ω)|2]. (5.26)We then simply have F [Γ(τ)](ω) = |s0(ω)|2.However, such spectral power is not injected, in full, into the molecularexcitation, if only a finite duration of the incoherent field is to be used inconducting a finite-time wave packet simulation. We therefore pre-emptivelytake a “sample” of this source field by modulating E0(t) with a well-behaved,long-duration function (t). The resulting field used for molecular excita-tion is then E(t) = (t)E0(t). Since now the duration of E(t) is finite, wecan Fourier transform the field in time [95] to calculate its power spectrum,which, when averaged over many field realizations, converges to a constantfraction of |s0(ω)|2. In turn, we can use it to construct a coherent pulse bycarefully choosing appropriate spectral phases. This way, any further com-parison between coherent and incoherent dynamics will be on equal footingin terms of total optical energy input.Figure 5.7 demonstrates the numerical implementation of such considera-tion, where s0(ω) and (t) are both assumed to be Gaussian. The modulation(t) has negligible effects on changing the shape of the ensemble-averaged985.2. Vibrational resonances excited by incoherent light0.00e+001.00e-042.00e-043.00e-04 17850  17900  17950  18000  18050Power spectra (atomic unit)Excitation frequency (cm-1)absorption cross-section (arb. unit)coherent power spectrum (atomic unit)incoherent power spectrum  (100-realization average, atomic unit)-2.00e-06-1.50e-06-1.00e-06-5.00e-070.00e+005.00e-071.00e-061.50e-062.00e-06 0  2  4  6  8  10  12  14  16  18  20Electric field (atomic unit)Time (ps)-2e-06-1.5e-06-1e-06-5e-0705e-071e-061.5e-062e-06 0  2  4  6  8  10  12  14  16  18  20Electric field (atomic unit)Time (ps)00.00010.00020.0003 17850  17900  17950  18000  18050Power spectra (atomic unit)Excitation frequency (cm-1)absorption cross-section (arb. unit)coherent power spectrum (atomic unit)incoherent power spectrum  (100-realization aver ge, atomic unit)Figure 5.7: Plotted in the top panel are the coherent (blue) and incoherent(orange) electric fields in time, and the modulation (t) (red, scaled). A pairof resonances, indicated by the peaks in the φ0 → Y/B optical transitionmatrix element as a function of energy, in the 17900-18000cm−1 range, isplotted in the bottom panel (red). The power spectrum of the incoherentfield (orange) is obtained from averaging over 100 field realizations, whereasthat of the coherent field (blue), approximately 0.01614|s0(ω)|2, is simplyscaled down from the power spectrum of the incoherent source.995.2. Vibrational resonances excited by incoherent lightpower spectrum from |s0(ω)|2, since its large width in the time domainresults in an essentially δ-like spectrum comparing with |s0(ω)|2, therebymaking their convolution identical to |s0(ω)|2.5.2.4 Stationary wave packet and flux dynamicsAs described above, the vibrational wave packet dynamics can be propagatedusing equations (5.23) and (5.24), with the electric field having the form ofequation (5.25). For the molecular wave packets, the kinetic energy operatoris implemented using mid-point numerical differentiation (of second orderaccuracy). Transition dipole moments VY,g(R) and VB,g(R), non-adiabaticcoupling VB,Y (R), along with J = 1 effective potentials Wg,Y,B(R), are takenfrom ref. [111]. The effective potentials are further modified artificially tohave negative imaginary potentials at R = 23-25 Bohr as absorbing bound-aries (Fig. 5.6). This makes the numerical propagation of wave functionsfeasible over very long timescales. The vibrational wave packets, belongingto the three relevant electronic channels, are represented on a spatial gridwith 2100 points in steps of 0.01Bohr, in the R = 4-25Bohr range. The wavefunctions are then propagated in time with standard RK4 method, in stepsof 0.01fs, according to the Schro¨dinger’s equation.The electric field of the light is assumed to be linearly polarized, with thetime-dependent part E(t) represented as an array with half the step size ofthe propagation (i.e. 0.005fs) due to the RK4 method. At each time point,the field isE(t) = (t)∑j[s0(ωj)∆ω]e−i[ωj(t−10ps)+Φj ] + c.c.(t) = exp[−(t− 10ps)2/2(3ps)2]s0(ωj) = s0 exp[− (ωj − 17950cm−1)2/(30cm−1)2]s0∆ω = [(103W/cm2)/0c]1/2 (5.27)where 2000 frequency components are taken from 17850cm−1 to 18050cm−1in ∆ω=0.1cm−1 intervals. The phase angles Φj are independent randomvariables uniformly distributed in [−pi, pi]. The Fourier transform of the1005.2. Vibrational resonances excited by incoherent lightGaussian envelope function (t) has a spectral width about 1.756cm−1.The spatio-temporal plots for the magnitude-squared wave functions areproduced according to above numerical specifications in Fig. 5.8. There is aclear, visual evidence of the presence and absence of vibrational coherence.Under coherent excitation, the motion of the wave packet is oscillatory. Theinitial excited state wave function is localized due to the very short coherentexcitation of the localized ground state φ0. Then after the pulse, it movesoutward due to the repulsive forces from both channels, and bifurcates intoa dissociative portion into the open channel and a localized portion thatbounces back. It then completes another period of oscillation, determinedby the energy separation between the resonances, before again transitioningpartly into the open channel.Seen in drastic contrast is the wave packet evolution for the incoher-ent field, averaged over 100 field realizations. The incoherent wave packethas a much more stationary form that, in time, only scales uniformly inmagnitude, but exhibiting no oscillatory motion. This is a clear sign thatvibrational eigenstates are excited, with no correlation between relativephases. To confirm this claim, also considered is the photoexcitation by near-monochromatic fields (Figure 5.8), simulating each individual frequencycomponents of the incoherent field. Its spatio-temporal plot, as a sum oversquared-magnitudes of individual component field excitations, bears extremeresemblance to the incoherent case.It’s then clear that, under such spectrally incoherent photoexcitation, noquantum phase correlation exists between eigenstates of the system, therebyeliminating any vibrational or electronic quantum coherence, as supportedalso by our previous study. However, this does not mean that there areno quantum dynamics under incoherent excitation. One particular time-dependent quantity that still reflects the photoexcitation dynamics is thelong-range vibrational flux for the wave packet in the open channel. Itrepresents in general the rate of probability, and in our case the relativeamount of molecules photo-dissociated, flowing across the area of the spher-ical shell with radius R along the radial direction. It is defined in quantum1015.2. Vibrational resonances excited by incoherent lightCoherentIncoherent(100-realization average)Sum of frequency componentsCoherentIncoherent(100-realization average)Sum of frequency componentsCoherentIncoherent(100-realization average)Sum of frequency components04e-138e-131.2e-121.6e-12 4  6  8  10  12  14  16  18  2004e-128e-121.2e-111.6e-11FluxTime (ps)Flux (atomic unit) coherent excitation (right axis)sum from all frequency componentsincoherent excitation  (100-realization average)04e-138e-131.2e-121.6e-12 4  6  8  10  12  14  16  18  2004e-128e-121.2e-111.6e-11FluxTime (ps)Flux (atomic unit) coherent excitation (right axis)sum from all frequency componentsincoherent excitation  (100-realization average)17900 1792517950 179751800005101520 02e-154e-156e-158e-15Flux (a.u.)Excitation  frequency (cm-1)Time (ps)01e-152e-153e-154e-155e-156e-157e-15Figure 5.8: Sum of the magnitude-squared wave functions in the Y and Bstates, i.e. |ψY (t, R)|2 + |ψB(t, R)|2, are plotted in the top three panels forthe relevant fields. They are plotted in the same spatial and temporal range,and the same colour scale. Vaguely shown is the extended portion for R > 8Bohr. But they are clearly reflected in the plot for the flux in the lower rightpanel. Middle panel is the flux as functions of time, created by componentfields of the form [s0(ω)∆ω](t)e−iωt, spaced by ∆ω = 0.1cm−1. The sumof the flux across the Excitation frequency axis produces the total flux (redcurve) in the panel below it.1025.2. Vibrational resonances excited by incoherent lightmechanics asFα(t, R) =~m Im[ψ∗α(t, R)ddRψα(t, R)]. (5.28)with the unit of probability per time, in electronic channel α. Such fluxis zero for a bound eigenstate, since the wave function would be real. Thecontinuum nature of the excited state of IBr, in this case, enables eigenstatedynamics under incoherent excitation. In particular, we choose R = 15Bohr, because it is much farther away from the bound regions and boththe Y and B potential curves are flat. In practice, one can measure suchdefined flux by examining, for example, the rate of photo-ionization signalat a particular solid angle, representing a fractional value of the above flux.The flux as a function of time for both types of electric fields are thenplotted in Figure 5.8, and reflects the corresponding wave packet dynamicsexactly. For the coherent case, since the oscillatory wave packet motionleaves a trail of “impressions” of its transition into the open channel, we seea train of periodic bursts of flux. Under spectrally incoherent excitation,the averaged flux follows the shape of the modulated field amplitude (t).When again summed over the near-monochromatic frequency componentfields, the total flux is seen to be the converging limit of the incoherentcase. Here we emphasize that all component fields contribute to the fluxdynamics. Namely, off-resonance frequency components, even though onlycausing significant flux temporarily, are also an important contributor to thetotal incoherent dynamics [113]. Furthermore, the flux contribution at eachindividual excitation frequency, as the middle panel of Figure 5.8 shows, ismuch higher near-resonance than off-resonance, at long times. The exactvalue may be determined by actual forms of the vibrational eigenfunctionsbetween the ground and excited states, or their Franck-Condon factors.5.2.5 ConclusionsIn summary, we have considered in this study the photoexcitation of amolecule into a pair of coupled electronic states of mixed bound-continuumnature, supporting both bound and dissociative vibrational motion. Theincident light can be a coherent pulse or its incoherent counterpart with1035.2. Vibrational resonances excited by incoherent lightonly spectral phases randomized, but sharing the same central frequency,bandwidth and power spectrum.We numerically computed the exact spatio-temporal evolution of theexcited-state quantum wave packet, and demonstrated that, when initiatedby the incoherent pulse, the molecule does not sustain the same quantum-coherent behaviours as that from coherent one. This is attributed entirely tothe loss of mutual quantum phase correlation, inherited from the incoherentproperty of the excitation light, between excited vibrational eigenstates. Asa result, the incoherent wave packet dynamics coincide with that formedvia summing the squared-magnitudes of wave packets, excited separately bynear-monochromatic excitations at all relevant frequencies included in thespectrum of the incoherent pulse.Furthermore, we confirm directly that incoherent light also initiates dy-namics, although not of coherent nature, where “movement” of the system isin the incoherent mixture of eigenstates, rather than coherent superpositionamong them. In the context of dissociating IBr molecule, the incoherent dy-namics is simply the flow of probability of finding the bond length at largevalues. In this case, one would not expect any coherent transfer of molecu-lar population or quantum superposition in general. We’d also like to pointout that if the continuum manifold of the Hilbert space considered here isprovided by a dissipative environment instead of the dissociative electronicstate, the incoherent dynamics will then represent a unidirectional heat flowinto the environment. Although without any quantum coherence, one cannevertheless speculate that the efficiency of such unidirectional flow of fluxcan be controlled, in principle, by incoherent means such as adjusting the“receiver” of the flux in energy so that resonances with greater flux hasstronger coupling and are better received. This calls for further research onthe resonant intramolecular energy or population transfer, after the inco-herent photoexcitation step.104Chapter 6Final conclusionIn this doctoral dissertation, a number of studies had been presented onthe interaction physics between light, from coherent and incoherent sources,and diatomic molecular systems. On a fundamental level, the electric fieldof the light, interacting with the transition dipole moment of the molecule,can bring the molecule to new quantum states consisting of different combi-nations of energy eigenstates. Due to the many internal degrees of freedomof even simple diatomic molecules, such transitions can be complex, espe-cially in the cases when the excitation field is strong in intensity and broad inspectral bandwidth. However, numerous mathematical formulations, deriva-tions and approximations can be made for us to gain useful insight into theunderlying mechanisms for these light-induced molecular transitions, andhow to actively control them via coherence properties such as interferencesand resonances. Particular innovative are the theories and practices of co-herent control techniques, where tuning only the coherent properties of theexcitation field can alter the outcome of the physical process.Such coherent control ability goes hand in hand with the coherence ofthe light source for the molecular excitation. Laser sources are excellentchoices by default, in terms of their intrinsic coherent characteristics such asamplitude and carrier-phase stability and spatial and temporal coherence;their bandwidth availabilities, ranging from narrowband CW-excitations, tobroadband, ultrafast femtosecond pulses; and also the accompanying tech-nologies that enable active manipulation of the laser pulses, such as chirpingand other general spectral shaping.Invariable to all studies in this thesis, we aim to achieve our research ob-jectives by first presenting a design for the excitation light field. When thefield is incident to a model molecular system, the resultant quantum dynam-105Chapter 6. Final conclusionics are analyzed, and would bring us to the confirmation of a hypothesizedoutcome, such as the high-efficiency transition to a particular target quan-tum state (e.g. ultracold molecular state), or a specific correlation betweenproperties of the light (e.g. intensity, coherence) and the characteristics ofthe resulting molecular state (e.g. linear response, stationary wave packets).Our main methodologies are mathematical analysis and numerical com-putation and simulation. We base on the quantum mechanical theories toformulate the quantum dynamics problem, provide exact or approximatesolutions, physical and analytical insights, or simplified derivations that aresuitable for computational analysis. When no simple or direct analyticalsolution can be derived, or randomness is required as part of the demon-stration, we conduct numerical calculations and simulations, to reveal thequantitative evolution and outcome of the molecular system.In the first study, we extend the general technique of stimulated Ramanadiabatic passage (STIRAP) to photoassociation processes, where collidingatoms are combined, under the action of a pair of coherent pulses, into deeplybound ground molecular state, thereby forming an ultracold molecule. Suchtransfer from atomic to molecular state is especially efficient when the atomsare at ultracold temperatures, where their extremely narrow kinetic energyuncertainty greatly enhances the adiabatic passage. However, the efficiencycan be additionally improved using atomic scattering resonances such asa Feshbach resonance. Previous studies have concluded that, based on theincreased eigenstate-to-eigenstate coupling (Franck-Condon factor) providedby the resonance, large resonance width is always preferred to increase theefficiency. However, by our computational analysis, we found that whenan ensemble of ultracold atoms is considered, the dynamical aspects of theresonance plays a crucial role in determining the best resonance width forhighest efficiency. With a narrow resonance, the effective collision rate, i.e.the number of atom pairs still in the collision process, is increased in the timewindow for photoassociation. This is because the collision duration betweenany pair of atoms are prolonged by a larger factor than the decrease inthe photoassociation probability. The net effect is then the existence of anoptimal, narrow resonance width for the best ensemble transfer efficiency.106Chapter 6. Final conclusionNext, we apply the coherent control strategy and demonstrate thatphase-only manipulation of a broadband, hight-intensity, ultrafast laser pulsecan give rise to linear molecular response to the field intensity. Althougha necessary relation with weak, perturbative fields, it has been assumed,mistakenly in most studies, to be absent in the strong-field domain. Ex-perimental observations where both linear response and coherent controlcoexist, then, relied on the above false assumption and lead to the con-clusion of phase-only coherent control using a perturbative excitation field,an impossibility according to conventional, closed-system coherent controltheories. Although open quantum systems, as later studies have shown,can indeed fill in the gap, we resolve this issue by providing pulse shapingscenarios where both essentially linear molecular response to high-intensitylight, and coherent control over electronic branching ratio, are present in iso-lated molecules. Examples using a model three-potential molecular system,and the realistic H+2 molecular ion, are provided. As the pulse manipula-tion is solely coherent, the reason for the existence of linear response in thestrong-field domain is then attributed to interference effects. Furthermore,wave packet interferometry analysis in the vibration of H+2 also reveals thecorrelation between the linear response and the wave packet’s concentrationoutside the Franck-Condon region. Meanwhile, control over branching ra-tios, and quantum beating in the population among various electronic androtational channels are also present.The last theme of this thesis is concerning the vibration of molecules un-der incoherence photoexcitation. With weak-intensity excitation light, suchas natural sunlight or laser light, a simple stochastic Schro¨dinger’s equationis derived for the excited vibrational wave packet dynamics. The temporalcoherence of the electric field is shown to directly govern the randomnessof the excited wave packet. Under the excitation of a field modelled aftera single thermal emitter, which suffers phase and frequency interruptions,the resultant vibrational wave packets are calculated for two overlappingelectronic potentials. The mutual electronic coherence is shown to be dras-tically decreased, comparing with the coherent counterpart of the field shar-ing the same spectral profile. Based on this, we conclude that sunlight,107Chapter 6. Final conclusionwhich consists of a large number of such thermally emitted fields at eachfrequency, can induce much less, if none, electronic coherence in molecules,as compared with that resulting from laboratory experiments using broad-band laser pulses. Therefore, although the long-lived quantum coherencediscovered in light-harvesting molecules, under laser excitation, is an inter-esting phenomena that holds many scientific and technological potentials,its relevance to natural photosynthetic processes is cast into great doubt.To further study the behaviour of molecular vibration under incoher-ent photoexcitation, we model another type of incoherent light source byintroducing randomness in the spectral domain, where each frequency com-ponent of the excitation has statistically random phase angles. When amolecule is excited with such field into a continuum of vibrational eigen-states, all frequencies of the light contribute to the state transition, and asa result, these vibrational eigenstates lose mutual coherence in the phasesas well. This causes excited vibrational wave packet that is stationary. Tomake the comparison clear, the excited region of the vibrational continuumalso hosts resonances, i.e. coupling to bound states that support boundedoscillatory motion. This enables oscillation of the wave packet created bythe spectrally coherent field, during its dissociation. Despite the lack of os-cillatory motion and quantum interference, incoherently excited vibrationalresonances nevertheless have a dynamical aspect: the outgoing probabilityflux at asymptotic distances. In the context of dissociating IBr molecule,such dynamics is the stream of dissociating atomic fragments. The flux val-ues near different vibrational resonances are also different, but always higherthan those off-resonance.The study on the incoherent excitation of vibrational resonances alone,then, calls for further investigations for the possibility of tuning into thestructural degrees of freedom of the molecule, in order to improve the effi-ciency of intra- or intermolecular transfer of population or energy. Specifi-cally, if the properties of the resonance, such as its peak location in energy,can be altered, the incoherent flux at a target energy can in principle becontrolled. One would then speculate that if the molecule has the abilityto change its own excited state PES, it can, under natural evolution, search108Chapter 6. Final conclusionfor the best shape of the PES in order to optimize flux or energy transfervia resonances. Such proposed approach can add motivation to improvethe understanding and possible manipulation of molecular processes such asintramolecular vibrational redistribution (IVR), intra- and intermolecularenergy transfer involved in photosynthesis, and the protein folding problem.On the more coherent side, the application of the coherent control ap-proach also sees endless possibilities. One particularly fascinating applica-tion is on the effects of coherent photoexcitation on mechanical motions ofa polyatomic molecule. The optical access to the electronic excitation ofa polyatomic molecule might mean, for example, the controlled breakingand re-forming of double bonds on a central part of the molecule. Differentbond structures can enable different torsional and other vibrational motions,hence opening up possibilities on controlling the mechanical motion of themolecule as a whole. Targeted and efficient quantum state transfer, coher-ent manipulation of vibrational wave packets, can all be excellent techniquesto achieve such goal. 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