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Ultrafast SpectroscopyQuantum information and wavepackets

Ultrafast SpectroscopyQuantum information and wavepackets

Joel Yuen-ZhouResearch Laboratory of Electronics, Massachusetts Institute of Technology

Jacob J KrichDepartment of Physics, University of Ottawa

Ivan KassalSchool of Physics and Mathematics, University of Queensland

Allan S JohnsonDepartment of Physics, Imperial College London

Alan Aspuru-GuzikDepartment of Chemistry and Chemical Biology, Harvard University

IOP Publishing, Bristol, UK

ª IOP Publishing Ltd 2014

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Permission to make use of IOP Publishing content other than as set out above may be sought atpermissions@iop.org.

ISBN 978-0-750-31062-8 (ebook)ISBN 978-0-750-31063-5 (print)

DOI 10.1088/978-0-750-31062-8

Version: 20140801

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Published by IOP Publishing, wholly owned by The Institute of Physics, London

IOP Publishing, Temple Circus, Temple Way, Bristol, BS1 6HG, UK

US Office: IOP Publishing, Inc., 190 North Independence Mall West, Suite 601, Philadelphia,PA 19106, USA

Contents

The authors vii

Glossary of common terms ix

Introduction x

1 The process matrix and how to determine it: quantumprocess tomography

1-1

Bibliography 1-7

2 Model systems and energy scales 2-1

Bibliography 2-7

3 Interaction of light pulses with ensembles of chromophores:polarization gratings

3-1

3.1 Laser-induced polarization gratings 3-1

3.2 Induced linear and nonlinear polarization in an ideal coupled dimer 3-3

3.2.1 Eigenvalues, eigenvectors, and energy scales 3-4

3.2.2 Induced polarization 3-5

3.2.3 Time and energy scales in the model 3-14

3.3 Measuring the signal: connecting induced polarization toexperimental results

3-15

Bibliography 3-19

4 Interaction of light pulses with ensembles of chromophores:wavepackets

4-1

4.1 Linear absorption spectroscopy 4-1

4.2 Pump–probe (PP0) spectroscopy 4-7

Bibliography 4-21

5 Putting it all together: quantum process tomography andpump–probe spectroscopies

5-1

5.1 Broadband PP0 spectra in terms of the process matrix 5-1

5.2 Performing QPT using PP0 data 5-7

Bibliography 5-22

v

6 Computational methods for spectroscopy simulations 6-1

6.1 Propagation of wavefunctions 6-1

6.2 Numerical simulation of frequency-resolved linear absorption 6-2

6.3 Numerical simulation of frequency-integrated linear andnonlinear spectra

6-7

6.4 Extensions: boundary conditions and relaxation dynamics 6-14

Bibliography 6-15

7 Conclusions 7-1

Bibliography 7-2

Appendices

A Mathematical description of a short pulse of light A-1

B Validity of time-dependent perturbation theory in thetreatment of light–matter interaction

B-1

Bibliography B-3

C Many-molecule quantum states of an ensemble ofchromophores interacting with coherent light

C-1

Bibliography C-6

D Frequency-resolved spectroscopy D-1

Bibliography D-9

E Two-dimensional spectroscopy E-1

Bibliography E-8

F Isotropic averaging of signals F-1

Bibliography F-6

Ultrafast Spectroscopy

vi

The authors

Joel Yuen-Zhou

Joel Yuen-Zhou is currently the Robert J Silbey PostdoctoralFellow in the Center of Excitonics at the Massachusetts Institute ofTechnology. He received a BSc in Chemistry and Mathematicsfrom the same school in 2007 and a PhD in Chemical Physicsfrom Harvard University in 2012. Starting in July 2015, he willbe an assistant professor in the Department of Chemistry andBiochemistry at the University of California San Diego. Hisresearch interests are broadly located in the realm of quantum

dynamics, specializing in nonlinear spectroscopy, quantum information,time-dependent density functional theory, and topological phases of matter.

Jacob J Krich

Jacob Krich is an assistant professor in the Department of Physicsat the University of Ottawa. He received his BA in Physics fromSwarthmore College in Pennsylvania, followed by an MMath fromOxford University, where he was a Rhodes Scholar. He receivedhis PhD in theoretical condensed-matter physics from HarvardUniversity. After receiving his PhD, Jacob was a Ziff Fellow of theHarvard University Center for the Environment and a postdoctoralfellow in the Department of Chemistry and Chemical Biology at

Harvard. His research focuses on novel pathways to high efficiency photovoltaicsand nonlinear spectroscopies of organic systems.

Ivan Kassal

Ivan Kassal is an ARC DECRA Research Fellow in the School ofMathematics and Physics at the University of Queensland. Hereceived his BS in chemistry fromStanfordUniversity in 2006 and hisPhD in chemical physics from Harvard University in 2010. He haspublished a number of papers on exciton transport in photosynthesisand on the application of quantum computers to problems inchemistry. His homepage is at http://www.ivankassal.com.

Allan S Johnson

Allan Johnson is a Marie-Curie Early Stage Researcher andNSERC PGS award holder in the Quantum Optics and LaserScience division at Imperial College London. Previously he obtainedhis undergraduate degree in Physics–Mathematics at the Universityof Ottawa. Previous to that, he was an actor and set designer. Hisresearch interests include nonlinear and nonperturbative quantumdynamics in the presence of strong and ultrafast laser fields.

vii

Alán Aspuru-Guzik

Alán Aspuru-Guzik is a Professor at Harvard University in theDepartment of Chemistry and Chemical Biology. He received hisdoctoral degree from the University of California, Berkeley, andwas the recipient of the 35 Innovators under 35 by MIT Technol-ogy Review for his contributions to the intersection of quantuminformation and quantum chemistry. In this context, he is inter-ested in how chemical experiments such as ultrafast spectroscopycan be interpreted using ideas from quantum information. More

about his work can be found on his home page, http://aspuru.chem.harvard.edu.

Ultrafast Spectroscopy

viii

Glossary of common terms

2D-ES Two-dimensional electronic spectrumB Bathχ Process matrix (equation (1.5))DEM Doubly-excited manifoldDS-FD Double-sided-Feynman diagramɛn Term in n-th pulse proportional to e�iωnt (ωn > 0, equation (3.4))

In the RWA, it promotes ket (bra) amplitude from lower (higher) to higher (lower)energy states (equation (3.25))

ɛ*n Term in n-th pulse proportional to eiωnt

In the RWA, it promotes (ket) bra amplitude from lower (higher) to higher (lower)energy states (equation (3.25))

ESA Excited-state absorptionFC Franck-CondonGSB Ground-state bleachGSR Ground-state recoveryGSM Ground-state manifoldH0(R) Molecular Hamiltonian as a function of nuclear coordinates R in the absence of

pulses (equation (2.4))LO Local oscillatorμ Dipole operator (equation (2.9))OQS Open quantum systemPC Phase cyclingPE Photon echoPES Potential energy surfacePP0 Pump probeQPT Quantum process tomographyRWA Rotating wave approximation (equation (3.25))S Systemσ Pulse duration (equation (3.4))η Light pulse electric field strength (equation (3.4))SE Stimulated emissionSEM Singly excited manifoldTG Transient gratingΩn

ij Transition amplitude for ket from j ji to jii or for bra from jii to jji via ɛn.In the RWA, it is significant only if ωij > 0 (equation (3.20))

Ωnji Transition amplitude for ket from jii to j ji or for bra from j ji to jii via ɛ*n:

In the RWA, it is significant only if ωij > 0 (equation (3.20))

ix

Introduction

Ultrafast spectroscopy is a powerful tool for studying excited-state processes inphysical systems ranging from atoms and molecules in the gas phase to condensedphases such as proteins, membranes, or solids. In a typical spectroscopic experiment,the system of interest is subjected to a series of incoming optical pulses, which triggernonequilibrium dynamics that are probed by the outgoing light, which constitutesthe signal. The adjective ‘ultrafast’ refers to the timescale of the phenomena ofinterest, which can span several orders of magnitude, from attoseconds to nano-seconds, and corresponds also to the timescales associated with the duration of thepulses and the separations between them. The information contained in the signal isone of the few windows we have to understand processes occurring at the nanoscale.Hence, the understanding of such techniques is essential to a wide range of modernresearch in physics, chemistry, and materials science. There is a certain art to thedesign of a spectroscopic experiment, and experienced practitioners will often comeup with creative ways to craft the optical pulses so that the signal reflects theinformation they are searching for. The theory of ultrafast spectroscopy, however,may be a daunting and confusing exercise for newcomers to the field, as it ofteninvolves mastering a formalism that ranges from electromagnetism to time-dependentquantum mechanics, and depending on the system of interest, to solid state physics,open quantum systems, molecular dynamics, and so on.

This text grew out of our efforts to understand ultrafast spectroscopy in aphysically intuitive way and from several different perspectives. One such perspec-tive uses concepts from quantum information theory and open quantum systems,which reframe the spectroscopic enterprise as a quantum process tomography(QPT), a protocol for systematically extracting the maximum amount of informa-tion possible from a quantum ‘black box.’ For our purposes, this box correspondsto the excited-state dynamics of the physical system of interest. Spectroscopy as aQPT then becomes an exercise in preparation and detection of quantum states. Thispersective gives insight into the sometimes convoluted perturbation-theory calcu-lations. This approach also provides limits on the amount of (quantum) informationthat a particular spectroscopic signal contains. The other perspective is that ofwavepacket dynamics, a ‘chemical’ approach that focuses on tracking the time-dependent quantum state of nuclear degrees of freedom along various electronicpotential energy surfaces. The strengths of this perspective are the great physicalinsights that can be obtained via intuition rooted in classical and semiclassicalmechanics, the computational advantages associated with their numerical imple-mentation, and the visualization opportunities associated with their simulation. Wehave also sought to provide a single source where one could find answers to ques-tions that were supposedly well-known but scattered over the literature.

This book should be accessible to anyone who has taken a full-year graduate-levelcourse in quantum mechanics, and is recommended to beginning researchers—theorists and experimentalists alike—who are interested in quantum dynamics andits experimental observables. We believe that established researchers will also obtain

x

novel physical insights on possibly familiar topics, depending on whether they comefrom ultrafast spectroscopy, quantum optics, quantum information, or theoreticalchemistry, to mention a few possibilities. The text can also serve as a textbook forspecialized courses or workshops, or the examples and code can be adapted to bepart of problem sets and exercises. A great portion of the material has emerged fromour own recent studies, so the topics are timely material for current research. Thediscussion is sufficiently detailed to allow the reader to design and calculate resultsfor nonlinear optical experiments on toy models (i.e., the coupled dimer), which canbe readily generalized to more complex systems that may appear in a research setup.Computational simulations are important for most nontrivial systems, and weprovide detailed descriptions and downloadable code to simulate an array of spec-troscopic signals.

The structure of the book is as follows. Chapter 1 introduces the quantuminformation perspective and is devoted to understanding the process matrix in anopen quantum system (OQS), i.e., a system interacting with an environment or bath.Chapter 2 introduces the important model systems that we use for examplesthroughout the book. Chapters 3 and 4 introduce the interaction of light pulses withmolecular systems, focusing on how to predict experimental outcomes from com-mon spectroscopic experiments. We use the time-domain wavepacket approach tounderstand both linear and nonlinear spectroscopies. Chapter 5 shows how well-crafted ultrafast experiments can reconstruct the quantum process matrix for thesingly-excited states of model molecular systems. Finally, chapter 6 contains adetailed discussion of the numerical simulation of spectra from a time-dependentperspective and should be useful for readers who are looking to implement com-putationally the ideas in this book. Example MATLABs code can be downloadedon the book’s website. We conclude with chapter 7 by describing the value andinsights of rethinking spectroscopy in terms of quantum information processingconcepts. There are six appendices, which address the mathematical description of ashort pulse of light, the validity of time-dependent perturbation theory in opticalspectroscopy, the equivalence of the many non-interacting molecule calculation andthe single-molecule one, a primer on frequency-resolved linear and nonlinear spec-troscopy, an introduction to 2D spectroscopy, and finally, the procedure to evaluatethe isotropic average of a spectroscopic signal. In particular, appendices A, B, and Coffer our perspectives on ‘well-known’ subjects that are often not covered in ele-mentary introductions. These are our attempts to fill this gap in the literature.

The book is naturally biased in its selection of topics, and is far from a com-prehensive treatise for ultrafast spectroscopy (for the latter, we invite the reader toconsult the classic textbook by Shaul Mukamel, Principles of Nonlinear OpticalSpectroscopy, Oxford University Press 1999). First, we restrict ourselves to elec-tronic spectroscopy, although it is clear that most of the tools presented here can begeneralized to other energy scales. Second, the material is presented using the modelsystem of the coupled dimer, whereas many real systems are more complicated. Thismodel effectively demonstrates the key ideas of the spectroscopic techniques and canbe easily adapted to treat more complex systems. We emphasize the techniques offrequency-integrated pump–probe (PP0) and transient-grating (TG) spectroscopies;

Ultrafast Spectroscopy

xi

we give brief, self-contained introductions to frequency-resolved and two-dimensionalspectroscopies in the appendices. Even though there is presently a wealth of activity inthe field of multi-dimensional spectroscopy, these techniques are not significantlydifferent in spirit from their PP0 and TG counterparts, and can be readily understoodafter having a solid grasp on these techniques. Our aim is to offer what we believe isthe simplest introduction to the field that brings the reader up to speed in termsof current research topics, yet without sacrificing physical intuition. In this regard,many interesting subjects such as Raman spectroscopy, response theory, fieldquantization, optical activity, infrared spectroscopy, semiconductor optics, andapplications of 2D spectroscopy have been deliberately omitted. We refer the readerto the book by Mukamel for a survey of such subjects, as well as the texts ofMinhaeng Cho (Two Dimensional Optical Spectroscopy, CRC Press 2009) as well asMartin Zanni and Peter Hamm (Concepts and Methods of 2D Infrared Spectroscopy,Cambridge University Press 2011).

The structure of the book is amenable to a front-to-back reading, but readers withparticular interests may easily skip chapters as several of them are self-contained.For example, if one is solely interested in understanding nonlinear spectroscopy viathe wavepacket rather than the quantum information approach, chapters 2, 3, and 4will suffice. Also, chapter 6 can be simply used as a reference while exploring theexample MATLABs code.

A few years ago, we were amongst those confused researchers entering the field ofultrafast spectroscopy. This book compiles many of the insights we gatheredthroughout the process of doing research on the subject, either by reading textbooksand papers, attending summer schools and conferences, but also speaking withtheorists and experimentalists at a personal level. In particular, we wish toacknowledge many discussions with our colleagues Dylan Arias, Keith Nelson, andSemion Saikin. Our approach to the subject is heavily influenced by the teachings ofEric Heller as well as David Tannor’s text Introduction to Quantum Mechanics:A Time-dependent Perspective, University Science Books, 2007. JYZ and AAGacknowledge support from the US Department of Energy, Office of Science, Officeof Basic Energy Sciences under Award Number DESC0001088. JJK and AJacknowledge support from NSERC. IK was supported by the Australian ResearchCouncil, under projects DE140100433, CE110001013, and CE110001027. We aresincerely grateful for the patience and encouragement from IOP editors John Navasand Jacky Mucklow.

With this, we hope that our text will help the reader acquire an appreciation forthe inner workings of nonlinear spectroscopy and use those concepts in creativeresearch contexts. We welcome comments and errata at joelyuenzhou@gmail.com.

The authorsCambridge, Ottawa, Brisbane, London, 2014

Ultrafast Spectroscopy

xii