NON-LINEAR OPTICAL SPECTROSCOPY (videos)
A complete English language introductory course of theoretical non-linear (optical) spectroscopy held at the Faculty of Mathematics and Physics of the Charles University as NOOE119: Non-linear Optical Spectroscopy (NOOE119: Nelineární optická spektroskopie). Non-linear optical spectroscopy is a crucial tool for observation of open quantum systems on the natural time-scales of their dynamics. However, the relation between observed spectroscopic signals and the quantum dynamics is highly non-trivial. This course is designed to shed light on this relation, and help students to find answers related not just to non-linear spectroscopy itself, but also to the general topics of open quantum systems, quantum biology and applied quantum mechanics.
Links
- The course website
- This course in the Student Information System
- Charles University courses at the 4EU+ European University Alliance webpage
- Charles University
- Faculty of Mathematics and Physics
- Lecturer: Tomáš Mančal (mancal at karlov.mff.cuni.cz)
Time and location of the course: any time, on-line (youtube channel)
Info, teasers, etc.
Description
Basic information about the lecture from the lecturer. This is the last time you see my face in the videos!
Videos and materials
Videos
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- Outline of the lecture (to be published)
Content
- 0:00:00 - Basic info about the lecture
Lecture 1: Introduction to non-linear spectroscopy
Description
Introduction to the course topic: What is non-linear spectroscopy, and how it is described by quantum mechanics. Relation of the experimental techniques to the perturbation theory Maxwell equations, electromagnetic potentials and electromagnetic theory of linear absorption.
Content
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- 0:00:00 - What is non-linear optical spectroscopy?
- 0:06:07 - Why non-linear spectroscopy?
- 0:09:30 - Macroscopic vs. microscopic observation
- 0:15:38 - Relation between spectroscopy and perturbation theory
- 0:21:03 - Example: Linear absorption
- 0:25:57 - Example: Pump-probe
- 0:31:32 - Molecules as open quantum systems, reduced description of quantum systems
- 0:43:39 - Maxwell equations and electromagnetic potentials
- 0:46:08 - Electromagnetic potentials
- 0:52:42 - Coulomb gauge
- 0:58:55 - Transverse and longitudinal fields
- 1:04:33 - Continuity equation, transverse and longitudinal currents
- 1:13:24 - Linear polarization and absorption, linear absorption coefficient
Lecture 2: N-wave mixing, dynamics of driven open quantum systems
Description
N-wave mixing as a non-linear light-matter interaction process. Introduction to the necessary tools, results and approximations of open quantum systems for non-linear optical spectroscopy.
Content
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- 0:00:00 - n-wave mixing (general description)
- 0:07:17 - Wave-equation and an ansatz for non-linear polarisation
- 0:19:34 - Solution of the wave-equation: ansatz and simplification
- 0:37:28 - Solution of the wave-equation: integration of the wave equation
- 0:44:05 - Intensity of the signal
- 0:46:08 - Phase matching condition
- 0:51:15 - Detection of non-linear signal (heterodyne detection)
- 0:54:50 - Pump-probe as a homodyne detected signal
- 1:02:55 - State vector of the relevant degrees of freedom
- 1:07:26 - Macroscopic polarisation through averaging and expectation values
- 1:10:32 - Total statistical operator and reduced statistical operator
- 1:20:22 - Equations of motion for the reduced statistical operator (intro)
Lecture 3: Equantions of motion for the reduced density matrix
Description
Reduced density matrix theory using super-operator notation. Evolution super-operator, Liouvillian and transition dipole moment super-operator. Phenomenological master equation to describe relaxation of the reduced density matrix. Secular approximation, dependence of the master equation on the initial condition.
Content
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- 0:00:00 - Brief summary - towards equations of motion with E = 0
- 0:05:59 - Useful tricks: Super-operator formalism (Liouvillian and evolution super-operator)
- 0:16:15 - Splitting the Hamiltonian of open quantum system
- 0:17:54 - Equation of motion in super-operator form, useful notation, evolution back in time
- 0:21:10 - Useful tricks: Interaction picture with super-operators
- 0:31:32 - Reduced density matrix from interaction picture with respect to bath
- 0:35:54 - Equation of motion for the reduced density matrix, relaxation super-operator
- 0:40:12 - Phenomenological theory for the relaxation super-operator
- 0:41:16 - Basis of state for the representation of relaxation super-operator
- 0:43:49 - Population transfer
- 0:48:30 - Decoherence and rephrasing
- 0:52:13 - Elements of the relaxation super-operator
- 1:00:12 - Secular approximation
- 1:03:32 - Dependence of equations of motion on the initial condition
- 1:08:54 - Brief summary
- 1:10:40 - Elements of evolution super-operator
- 1:14:42 - Summary before perturbation theory
- 1:17:30 - Transition dipole moment super-operator
Lecture 4: Time-dependent Perturbation Theory, Response Functions
Description
Time-dependent perturbation theory in the classical electric field. The three important steps towards perturbation expansion of the equations of motion, definition of the time ordered exponential and detailed discussion of its n-th order. Definition of the response function and an example in the third order. Level structure of the investigated molecular systems. Response function of a two-level system consists of eight terms.
Content
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- 0:00:00 - intro and definitions
- 0:05:43 - time-dependent perturbation theory (TD-PT) with respect to E
- 0:07:50 - three steps towards TD-TP
- 0:08:21 - 1) interaction picture
- 0:14:33 - 2) integration
- 0:16:09 - 3) iteration
- 0:24:20 - Time-orderred exponential
- 0:26:57 - n-th order of the exponential
- 0:29:08 - n-th order of the polarization
- 0:34:59 - 3rd order polarization
- 0:50:48 - simplification using equilibrium properties
- 0:55:42 - definition of the response function
- 0:56:53 - structure of the response function: in time
- 1:04:49 - structure of the response function: in Liouville space
- 1:07:38 - structure of the V superoperator (level structure, transition dipole operator)
- 1:15:02 - 1. interaction
- 1:16:43 - 1. time evolution - coherence evolution
- 1:22:48 - response function of a two-level system
Lecture 5: Liouville pathways, three pulse experiment
Description
Double-sided Feynman diagrams for Liouville pathways. Rephasing and non-rephasing pathways and selection of the signal type by signal direction. General arrangement of a three-pulse experiment. Dependence of the signal type on pulse ordering.
Content
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- 0:00:00 - Short summary
- 0:04:56 - Double-sided Feynman diagrams
- 0:13:42 - Positive sign diagrams
- 0:18:31 - Negative sign (complex conjugated) diagrams
- 0:19:51 - Standard notation - R1, ..., R4
- 0:23:39 - Graphical representation of Liouville pathways
- 0:27:26 - Phase factors of the Louville pathways
- 0:37:39 - All pathways with phase-factors
- 0:41:45 - Rephasing and non-rephasing phase factors/pathways
- 0:45:20 - Summary on polarization, selected single contributions to the polarization
- 0:49:07 - General three pulse arrangement
- 0:53:07 - Electric field representing three pulses
- 0:57:48 - Selection of the signal k-vector
- 0:59:45 - Pulse order 1-2-3
- 1:10:02 - Resonant excitation condition
- 1:12:04 - Ultra-short pulse condition
- 1:17:43 - Only rephasing pathways contribute to -k1+k2+k3 and 1-2-3
- 1:21.36 - Pulse order 1-2-3, direction k1-k2+k3: non-rephasing signal
- 1:27:34 - Pulse order 2-1-3, direction -k1+k2+k3: non-rephasing signal
Lecture 6: Geometry of the three-pulse experiment, spectroscopic lineshapes, heterodyne detection
Description
Geometry of the three pulse experiment and separate detection of rephasing and non-rephasing pathways by spatial and temporal arrangements of pulses. Introduction to the interaction ordering problem with pulses of finite length. Relations of Liouville pathways to absorption lineshapes. Lineshapes as components of Liouville pathways. Heterodyne detection of non-linear response and the problem of phase. Linear absorption as homodyne detection of the first order signal.
Content
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- 0:00:00 - Geometry of the three-pulse experiment
- 0:13:35 - Problems with interaction ordering and limited spectral width of finite pulses
- 0:22:31 - Linear absorption using first order Liouville pathways
- 0:46:35 - Complex lineshape
- 0:49:39 - Liouville pathway with lineshapes as its components
- 0:55:42 - Homodyne detection of the non-linear signal field
- 1:01:35 - Influence of phase uncertainty on detected line shape
- 1:08:41 - Absorption as homodyne detection of first order field
Lecture 7: Pump-probe spectroscopy (part 1)
Description
Pump-probe experiment as a differential absorption spectroscopy in phenomenological formulation. Discussion of Liouville pathways involved in the pump-probe non-linear response. Expression of Liouville pathway contributions in terms of absorption line shape. Formulation of the response using evolution superoperator elements for the excited state evolution. Two-level system with excited state decay. Derivation of the evolution superoperator for the excited state decay.
Content
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- 0:00:00 - Pump-probe as a third order absorption
- 0:19:16 - Pump-probe as a differential absorption (conventional sign of the absorption)
- 0:23:57 - Pump-probe signal in terms of individula Liouville pathways
- 0:33:43 - Liouville pathways in terms of absorption line shapes
- 0:40:30 - Pump-probe of a two-level system
- 0:45:12 - Liouville pathways in pump-probe
- 0:53:00 - Indistinguishability of rephasing and non-rephasing signals in pump-probe
- 0:56:59 - Stimulated emission and ground-state bleach
- 1:01:13 - Two-level system with decaying excited state
- 1:11:11 - Evolution superoperator elements as conditional probabilities
- 1:16:42 - Evolution superoperator element for excitate state decay
Lecture 8: Pump-probe spectroscopy (part 2)
Description
Liouville pathways for energy transfer processes. Identification Liouville pathway types for energy transfer pathways. Complete list of Liouville pathways for a relaxing two-level system. Differential absorption (DA) in terms of Liouville pathways. General expression for pump-probe (PP) also known as DA and its dependence on time T. Response of the systems with excited state absorption. Generalization of the theory to a multi-level, multi-band systems. Recipe how to calculate absorption and DA spectra. Introduction to the problem of orientational averaging in an isotropic sample.
Content
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- 0:00:00 - Liouville pathways with energy transfer processes
- 0:09:16 - Identification of Liouville pathway type for energy transfer pathways
- 0:15:58 - List of Liouville pathways for a relaxing two-level system
- 0:22:26 - Differential absorption (Pump-probe) in terms of Liouville pathways
- 0:24:49 - Differential absorption at T=0
- 0:28:14 - Differential absorption at general time T, pump-probe measurement of a excitation decay process
- 0:33:50 - Systems with excited state absorption
- 0:35:23 - Liouville pathways of excited state absorption
- 0:42:58 - Generalization of the theory to multi-level systems
- 0:50:52 - How to calculate absorption spectrum (complete expression)
- 0:56:18 - How to calculate pump-probe (differential absorption) spectrum (complete expression)
- 1:10:04 - Rotational averages over transition dipole moment directions (intro)
- 1:18:45 - Orientational average of the dipole factor in absorption
Lecture 9: Orientational averaging in pump-probe and magic angle
Description
Rotational averaging of the third order Liouville pathways. Examples for pathways describing population decay, energy transfer and excited state coherence. Derivation of the magic angle condition for a single transition and in a general case. Cancelation of the signal of even order spectroscopies in an isotropic sample. Observation of energy transfer by pump-probe spectroscopy and construction of two-dimensional frequency resolved pump-probe.
Content
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- 0:00:00 - Rotational averaging in third order Liouville pathways
- 0:10:23 - Examples of averaged prefactors for particular Liouville pathways
- 0:40:06 - Magic angle, derivation for a single transition
- 0:54:57 - Magic angle for a general case
- 1:07:00 - Even order spectroscopies have a zero signal in isotropic sample
- 1:13:03 - Monitoring energy transfer by pump-probe spectroscopy, two-dimensional pump-probe
Lecture 10: Pump-probe spectroscopy with pulse of finite temporal width
Description
Discussion of the spectrally narrow excitation pulses and the trade-off between time and frequency resolution in pump-probe spectroscopy. Formulation of the non-linear response with finite pump-pulses and infinitely broad probe pulse. Convolution theorem and its application in evaluation of the response. Example of narrow line shapes and a formula for correction of the pump-probe spectrum to pump pulse spectrum.
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- 0:00:00 - Spectrally narrow pump, trade-off between frequency and time resolution
- 0:04:51 - Pump-probe with pump pulses of finite duration (formulation of the problem)
- 0:7:22 - Third order polarization with finite pump
- 0:9:33 - Adding the infinitely broad probe pulse condition
- 0:30:51 - Identification of convolution and the convolution theorem
- 0:33:37 - Application of the convolution theorem to the complex lineshape
- 0:45:59 - Preliminary meaning of the lineshape - pulse shape overlap
- 0:55:56 - Example of narrow pulse shapes
- 1:18:40 - Erratum
Lecture 11: Construction of spectroscopy with simultaneous time and frequency resolutions
Description
Details of the pump-probe detection scheme and its generalization to two non-colinear pump pulses. The idea of a generalized susceptibility for the probe pulse gets scrutinized and found valid under relatively general conditions. Constructing spectrum from complex lineshapes and the definition of a coherent two-dimensional (2D) spectra, which could provide a simultaneous high resolution in time and frequency. Basic properties of 2D spectrum.
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- 0:00:00 - Detection of pump-probe signal (once again), generalized susceptibility
- 0:10:01 - Generalization of the detection scheme to three non-colinear pulses
- 0:13:22 - Heterodyne detection of the signal using the local oscillator
- 0:20:20 - Third order polarization using Fourier transformed response function
- 0:50:36 - Third order polarization in frequency domain
- 0:55:31 - Conditions of "existence" of the generalized susceptibility
- 1:04:45 - Evaluation of the generalized susceptibility
- 1:16:35 - Constructing spectra from Liouville pathways; complex line shapes
- 1:21:23 - Opportunities opened by detection of complex signals
- 1:30:36 - Two-dimensional (2D) coherent spectrum (definition)
- 1:31:53 - Basic properties of the 2D spectrum
Lecture 12: Two-dimensional Cohetent Fourier-transformed Spectroscopy
Description
Summary of the expected properties of 2D spectrum introduced earlier. Geometric and temporal arrangements of the pulses in the 2D experiment. Detection of the complex non-linear signal and determination of its unknown phase factor. Example spectrum of a donor-acceptor system exhibiting energy transfer and no excited state delocalization. Complete discussion of the Liouville pathways composing the non-linear signal. Placement of the contribution of the individual Liouville pathways into a 2D spectrum.
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- 0:00:00 - Summary of two-dimensional spectrum
- 0:07:01 - Geometric arrangement of the experiment
- 0:14:31 - Temporal arrangement of the experiment
- 0:17:49 - Detection of the signal
- 0:21:30 - Determination of the complex signal
- 0:36:23 - Fixing the phase factor - projection relation to pump-probe
- 0:49:50 - 2D spectrum of an energy donor-acceptor system
- 0:51:36 - Collective of the donor-acceptor system
- 0:53:43 - Excited state evolution superoperator elements
- 0:56:00 - Liouville pathways through the system
- 1:24:44 - Placement of the pathway contributions
- 1:35:27 - Amplitude of the signal contributions (intro)
Lecture 13: Properties of Two-dimensional Spectra
Description
Amplitudes of the peaks in the 2D spectrum. How is disorder in molecular transition energies influencing absorption and 2DFT spectra. Additivity of the 2DFT spectrum for an ensemble of molecules and the identification of homogeneous and inhomogeneous linewidths in 2DFT spectrum. Energy transfer and excitonic interactions in 2DFT spectra. Summary of 2DFT standard properties. Application of 2DFT on observation of electronic coherence.
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- 0:00:00 - Amplitude of the signal contributions
- 0:09:01 - Energetic disorder in absorption and 2DFT
- 0:012:23 - Homogeneous and inhomogeneous broadening
- 0:18:33 - Additivity of ensemble spectrum
- 0:21:26 - 2DFT of a system with disorder
- 0:25:42 - Inhomogeneity in energy transfer
- 0:30:33 - Excitonic interaction in 2DFT spectroscopy
- 0:49:15 - Transition dipole moments of excitonic transitions
- 0:50:47 - Excitonic interaction in 2DFT spectroscopy
- 0:56:18 - Amplitude of the 2DFT crosspeaks at T=0
- 1:00:06 - Summary of 2DFT properties
- 1:03:13 - Electronic coherence in 2DFT spectrum
Lecture 14: Overview and Summary
Description
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BONUSES
Bonus 1 (Lecture 15): Colinear detection of non-linear signal - phase cycling
Description
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- Lecture 14 (in preparation)
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Bonus 2 (Lecture 16): Lineshape Dynamics - Environment Reorganization Effects in Spectroscopy
Description
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- Lecture 15 (in preparation)
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- 0:00:00 -