Stimulated Raman Scattering in albumin

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MASTERTHESIS INPHYSICS

Ultrafast spectroscopy and coherent control of

Tryptophan-based compounds

Author:

Luana Olivieri

Supervisor:

Prof. Tullio Scopigno

Co-Supervisors:

Dr. Giovanni Batignani Dr. Luigi Bonacina

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iii

Acknowledgements

Prima di tutto vorrei esprimere la mia gratitudine ai miei relatori il Prof. Tullio Scopigno e il Prof. Jean-Pierre Wolf che mi hanno dato la possibilità di partecipare a questo progetto.

Un profondo e sincero ringraziamento va al Dott. Luigi Bonacina e al Dott. Julien Gateau che mi hanno guidato durante l’esperienza a Ginevra, insegnandomi molto dal punto di vista sia pratico che teorico sul controllo coerente di sistemi quantistici. Allo stesso modo, un sentito ringraziamento va al Dott. Giovanni Batignani per il suo aiuto e supporto negli esperimenti di pump-probe (e per la sua infinita disponibilità).

Vorrei poi ringraziare Elise Schubert, Michel Moret, Denis Mongin, Va-syl Kilin, Nicolas Berti, Gustavo Sousa, Gabriel Campargue, Carino Fer-rante e Alessandra Virga per aver creato una stimolante e confortevole at-mosfera lavorativa.

Sono particolarmente debitrice alla mia famiglia e a tutti i miei amici e colleghi. In particolare vorrei ringraziare Giuseppe Bellanti per l’ imperit-uro supporto e il costante incoraggiamento profuso che mi ha sostenuto nei momenti più difficili di questa esperienza.

In fine, vorrei ringraziare l’università "La Sapienza" e l’università di Ginevra per avermi finanziato quest’esperienza attraverso la borsa di stu-dio "SEMP".

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v

Introduction

The chance to observe and manipulate ultrafast dynamics is an intrigu-ing goal in physics and chemistry. The development of femtosecond laser sources in 60’s paved the way to the realization of spectroscopic pump-prove techniques able to track electronic reconfiguration and visualize struc-tural rearrangements in the femtosecond time scale.

More recently, a fascinating perspective in the ultrafast community is to ob-tain an active control over complex systems dynamics. Processes like the breaking or the formation of specific chemical bonds are often hampered by the rapid redistribution through all the molecules of the energy locally deposited by mean of a femtosecond laser source, causing a leak of the se-lectivity. Thereby, the field of quantum coherent control emerged from the goal to drive a quantum system from an initial state to a desired final state by exploiting constructive quantum-mechanical interferences: it gives the chance to enhance the transition amplitude of a selected final state and at the same time exploits destructive interference to suppress undesired final states. The initial ideas exploited phase-controlled laser fields to manipu-late quantum-mechanical phases, as proposed by Brumer and Shapiro [1], or precisely timed sequences of ultrashort pulses, as proposed by Tannor and Rice [2,3]. In 1992, H. Rabitz and coworkers introduced the concept of optimal control, in their seminal paper "Teaching laser to control molecules" [4]. They proposed to use a search algorithm to optimize the laser pulse characteristics in a feedback loop configuration to reach most efficiently the desired target.

Thus today, a large number of parameters (such as the amplitude and the phase of each spectral component within the laser pulse) has to be con-trolled with new pulse shaping techniques and the help of efficient genetic-type optimization algorithms [5,6].

Within the context of coherent quantum control, quantum manipulation techniques have been applied to unravel microscopic informations on the system and discriminate between different, but very similar, compounds. In this respect, Optimal Dynamic Discrimination (ODD) is a powerful the-ory that is based on the enhancement or reduction of fluorescence of a spe-cific molecule by driving it preferentially into other relaxation pathways. It has been recently demonstrated experimentally achieving the discrimina-tion between small molecules like Riboflavin and Flavin mononucleotide,

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x Contents

Tryptophan and Tyrosine, Tryptophan and Ala-Tryptophan [7,8,9].

Even if the ODD theory allows measuring a target objective, which en-able the identification and discrimination between different compounds, there is not yet a procedure able to combine the informations obtained from the shaped pulse to the dynamics that take place in the excited state. Pump-probe techniques based on ultrafast transient absorption (TA) are able to extract these informations while non-linear Raman spectroscopies can un-veil the structural conformations of the system.

Transient absorption spectroscopy in picosecond and femtosecond time do-mains is a sensitive spectroscopic technique for studying the time evolution of excited states and the lifetimes of short-lived intermediates, and it is also useful to follow the energy flow among different chromophores composing the macromolecules.

The Stimulated Raman scattering (SRS) was one of the first nonlinear op-tical processes experimentally observed (1962) and its pump-probe version has recently been used to unveil the vibrational and rotational modes of liquids and gases [10].

Moreover, the combination of TA and SRS gives the chance to define a Fem-tosecond stimulated Raman Scattering (FSRS) experiment. FSRS is an ul-trafast nonlinear optical technique able to access the vibrational structural informations of the excited states combining high temporal precision and high spectral resolution within a three-pulse scheme experiment [11]. An actinic pump pulse excites the molecule and initiates the photochemical re-action. The transient structure of the molecule, represented by its Raman spectrum, is then visualized at various time delays by the combination of a narrowband Raman pump pulse and a broadband probe pulse.

This thesis addresses the study of the ultrafast dynamics of Tryptophan and Tryptophan-containing proteins within a Swiss NCCR - Molecular Ul-trafast Science Technology (MUST) project, thanks to the collaboration of two physics groups: the Femtoscopy group of Rome led by prof. Tullio Scopigno and GAP Biophotonics group of Geneve guided by prof. Jean-Pierre Wolf that was possible thanks to the Swiss-European Mobility Pro-gram (SEMP) internship.

This work aims to study the application of the optimal dynamics discrimi-nation theory to two proteins that contains Tryptophan, human serum albu-min (HSA) and immunoglobulin G (IgG), with the help of coherent control experimental apparatus. In fact, since its great sensitivity to environment, Tryptophan’s fluorescence can be the perfect probe for the detection of dif-ferent proteins and larger molecules [12] (see Chapter1). At the same time, TA is a powerful tool to unravel the dynamics of the excited states, while

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Contents xi

SRS experiments can unveil structural informations.

Chapter1points out the Tryptophan’s structure and the literature linked to its ultrafast dynamic, as well as human serum albumin and immunoglobu-lin G’s roles in human serum.

In Chapter2we define the coherent control and the ODD theories, outlin-ing the role of the Genetic Algorithm in the discrimination process, while Chapter3contains the transient absorption and the stimulated Raman scat-tering theories.

In Chapter4we describe the experimental apparatus used for all the mea-surements taken during this work: the setups employ in GAP Biophotonics lab. in Geneva and in Femtoscopy lab. in Rome. In particular, a SD FROG stage is mounted in the GAP’s setup and the software that controls the de-lay line and the CCD camera was built in Labwindows environment (see appendixA).

In Chapter5we investigate the time-resolved fluorescence of human serum albumin (HSA), Immunoglobulin G (IgG) and Trp. In particular, an ODD test is applied to discriminate between HSA and IgG in buffer solutions. Chapter6contains the investigation of the ground state normal modes and the excited states of Trp, made through the ultrafast transient absorption and a stimulate Raman scattering measurements.

In chapter7we present the conclusion of our work indicating possible fu-ture developments in the study of Tryptophan-based compounds.

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1

Chapter 1

Tryptophan

Tryptophan (Trp) is one of over 20 amino-acids which made up the pri-mary structure of proteins and usually dominates their fluorescence. Since Trp’s fluorescence is strongly influenced by its local environment, it can be often exploited to investigate the protein structure, conformation, fold-ing/unfolding, substrate binding and protein-protein interactions [13,14]. The structure of Trp is shown in figure 1.1: its side chain is composed of an indole, which is a chromophore responsible for most of the UV absorp-tion and fluorescence in proteins.

(A) (B)

FIGURE1.1: Structure of Tryptophan

(A) the bidimensional structure; (B) the 3D structure. Both the fig-ures are taken fromhttp://www.chemspider.com.

The indole’s excited state 1ππ∗ is characterized by two low-lying, nearly degenerate singlet states responsible for the absorption band centered at 280 nm (see figure 5.7b) [15, 16]. 1La and1Lb absorption bands are

pop-ulated through transitions among the highest occupied molecular orbitals (HOMO) and lowest unoccupied molecular orbitals (LUMO)

1_L

a:

HOMO → LUMO

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2 Chapter 1. Tryptophan in a ration 3:1 ; 1_L b : HOMO − 1 → LUMO HOMO → LUMO + 1

in an equal mixture [17]. Even if1Lb has lower energy and dominates the

emission in a nonpolar environment, the degeneration between the two ab-sorption band is removed in a polar solvent and a new energetic order is defined: the1La energy is decreased by interaction with the solvent, and

dominates the emission [18].

In addition, the absorption presents a band centered at 220 nm, named1Ba

and1Bb, that is due to higher intense transitions in the 190 nm - 230 nm

re-gion. They are characterized by greater energetic transitions from HOMO to higher lying LUMO, LUMO+1 and LUMO+2 [15].

(A) (B)

FIGURE1.2: (A) Indole potential energy profiles of the

low-est excited states1_ππ∗ _{(squares and diamonds), the lowest} 1

πσ∗ state (triangles) and the electronic ground state (cir-cles) as a function of the NH stretch reaction coordinate. From [19]. (B) Absorption spectrum of Trp in pH7 aque-ous solution. The absorption band at 280 nm is attributed to1Laand1Lb, while the absorption at 220 nm is due to1B

states [17,18,15]. Taken from [20]

1.0.1 Ultrafast dynamics of Tryptophan

The ultrafast dynamics of Tryptophan is generally linked to the quenching of the excited state S1 (usually related to1Lb absorption peaks) which

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Chapter 1. Tryptophan 3

[21,22].

In aqueous solution, besides sub-ps photoionization [23,24] and intersys-tem crossing [25,19,26] at long times (nanoseconds), the intermediate, pi-cosecond time scale is characterized by a multi-exponential fluorescence decay.

The picosecond time scale has been investigated by ultrafast transient absorption technique, and has been inquired the quenching of S1 excited

state ( with λ ∼ 320 nm) that has a characteristic time of 1.1 ps due to the solvent dipoles reorientations. Linked to the S1 quenching, an ESA band

centered at 360 nm rises simultaneously (see figure1.3) [27].

(A) (B)

FIGURE1.3:

(A): Time-resolved absorption spectra of tryptophan at pH 7, highlighting the formation of the 425 nm band due to the formation of P photoproduct and an energetic shift in the

300-400 nm range due to S1quenching.

(B): DADS obtained from global analysis of transient ab-sorption spectra from 300 to 550 nm. Positive and negative amplitudes denote respectively the decay and growth of transient absorption bands on the corresponding time

con-stant. Taken from [27].

Moreover, as outlined in figure1.3b, a new species rises at 425 nm with a characteristic time constant of 0.9 ns: it has been assigned to the primary P photoproduct, a triplet state termed T1, that results from the intramolecular

proton transfer from NH+₃ of zwitterionic form of Trp at pH 7.4 to the indole moiety (see figure1.4)[27,26,25].

The ESA bands for λ > 430 nm has been ascribed to the solvation process whose peak is estimated around 700-720 nm [24], while a peak due to Trp radical cations near 560-580 nm was observed only on the nanosecond time scale and in acid condition [27,28,25].

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4 Chapter 1. Tryptophan

FIGURE 1.4: Intramolecular proton transfer from NH+3 of

zwitterionic form of Trp to the indole moiety. Taken from [26].

1.0.2 Raman scattering on Tryptophan

Raman bands of Tryptophan residues are strong enough to be readily iden-tified in the spectra of proteins and their frequencies and intensities are con-sidered to reflect the environment of the residues [12]. The Raman spectrum of Tryptophan at a neutral buffer is reported in figure1.5: the intensity of the peaks are influenced by varying the Raman wavelength and reach their maximum at λ ∼ 200 − 220 nm (at resonant condition with the absorption band reported in figure1.2b) while decreasing at upper wavelengths [20,

29]. The most important normal modes are identified in table1.1and dis-cussed below.

FIGURE1.5: Resonance Raman spectra of aqueous

trypto-phan (1 mM) with 200-, 218-, 240-nm and 266-nm excitation. Numbers in parentheses correlate with benzene or pyrrole

mode (π) assignments [29].

The 1623 cm−1 Raman line (W1) is assignable to the highest frequency bond-stretching vibration of the indole ring [35], and with 1555 cm−1(W3) seem to come from the degenerate stretching Raman line of benzene at 1596

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Chapter 1. Tryptophan 5

Mode Frequency [cm−1] Mode Frequency [cm−1]

W1 a,b 1615-1622 W8b 1305 W2b 1575 W10a,b 1235-1238 W3 a,b 1552-1555 W13b 1127 W4b,e 1488-1496 W16a,b 1011-1016 W5 a,b 1457-1462 W17a,b 875-880 W6e 1420-1440 W18a,b 759-760 W7b 1360,1341 15 or δ NHc 1148 c 1256 5 πf,c 1276-1305 R’ φf 607 δCHπf 1064

TABLE1.1: Tryptophan mode’s frequency

a

: data are taken from [30];b: data are taken from [31];c: from [29];

d

: from [32];e: from [33];f: from [34].

cm−1. Normal mode W3 is also implicated to C-C stretching in the pyr-role nucleus [32] and its off-resonance intensity makes it a good marker for Tryptophan in proteins [36].

The 1436 and 1344 cm−1peaks (W6 and W7) can be linked respectively with 1480 and 1380 cm−1Raman lines of pyrrole . In particular, the normal mode W6 is mainly due to NH in-plane bending that shows a strong de-pendence upon the strength of hydrogen bonding [33]. Thus W6 can shift from 1422 (no hydrogen bonding) to 1441 cm−1(strong hydrogen bonding). A shift to 1382 cm−1has been reported in case of N-deuteration [35].

Harada and Miura’s works [32, 37] identify the origin of the doublet at 1360 and 1340 cm−1: they are assigned to Fermi resonance involving a fundamental (W7) expected near 1350 cm−1and one or more combination modes of out-of-plane vibrations. The relative intensity ration is a useful marker of the hydrophobicity of the ring. The tryptophan Fermi doublet is also observed in UVRR spectra of proteins [29].

The W17 normal mode at 880 cm−1involves both the deformation of the six-membered ring and displacement of the NH group along the direction of the N-H bond [32]. It is also a good marker for NH bond: from 883 cm−1 (no hydrogen bonding) to 871 cm−1 (strong hydrogen bonding) [37]. This component is detectable also in more complex tryptophan-based proteins [36].

The Raman line at 760 cm−1(W8) has been assigned to a mode in which breathing vibrations of benzene and pyrrole take place in phase.

Other components are detected in other articles [29,32,34] and recog-nized through the Wilson’s formalism for Benzene’s normal modes [38,39] and Lord’s formalism for Pyrrole’s normal mode [40].

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1.1 Tryptophan contained in proteins

Proteins are the building blocks of life since an arbitrary composition of them can lead to every living cells. They are polymers consisting of at least one sequence of amino-acids linked together in a polypeptide chain. Each amino-acid has its own position in the protein, dictated by the nucleotide sequence of genes. Any small change or damage in the sequence can mis-lead the overall protein function and can bring about some dangerous dis-eases. Therefore, understanding how proteins work requires informations about their structures and dynamics.

The complex 3D structure of a protein has been investigated in the last decades with different techniques as X ray crystallography [41], Nuclear Magnetic Resonance (NMR) [42] and Circular Dichroism (CD) [43].

Futhermore,the development of femtosecond laser sources disclosed the way to ultrafast spectroscopy studies. In particular, it is grown the capabil-ity to follow in real time the structural and energetic changes of electronic and vibrational excited states.

In this part we focus on a few examples of serum proteins for their relevant role in diagnostics of diseases [44]. In particular our study revolves around the serum albumin protein, which is responsible for transport of various compounds through the blood vessels and Immunoglobulin G (IgG) which is a protein of the immune system, an antibody that defends the body against foreign substances.

1.1.1 Human Serum Albumin

Human serum albumin (HSA) is one of the most abundant protein in plasma and constitutes approximately half of the proteins found in human blood. This protein consists of 585 residues set in a single polypeptide chain stabi-lized by 17 disulfide links, with three R-helical domains (I-II-III), each one contains two subdomains A and B (figure1.6).

The crystal structure analyses indicate that the principal regions of ligand binding sites in albumin are situated in hydrophobic cavities placed in sub-domains IIA and IIIA. These binding sites are known as Sudlow I and Sud-low II, respectively, and the only tryptophan residue in HSA is located in Sudlow I (Trp-214) [45,46,47].

HSA plays an important role in transporting various types of endoge-nous and exogeendoge-nous compounds like fatty acids, ions, heavy metals, hor-mones, amino acids, drugs.

As a consequence, it is well known as a marker for good nutrition and health: e.g. a decrease in HSA concentration indicates a negative acute-phase marker of inflammation or infection.

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1.1. Tryptophan contained in proteins 7

FIGURE1.6: Human Serum Albumin’s structure

notice the presence of one tryptophan residue (TRP 214). Picture taken from [45].

important for the distribution of body fluids between intravascular com-partments and tissues that is also very crucial for the regulation of body temperature.

1.1.2 Immunoglobulin G

Antibodies, also know as Immunoglobulins (Ig), are the constituent pro-teins of the immune system whose task is the identification and neutral-ization of pathogens such as bacteria and viruses . There are 5 classes of immunoglobulins differing considerably in their structures and biological functions: IgG, IgE, IgM, IgD and IgA (figure1.7).

FIGURE1.7: The 5 classes of Immunoglobulins: G, E, M, A,

D.

A basic structure of an immunoglobulin is a Y-shaped molecule com-posed of two regions: an heavy chain ("H", coloured in blue) that consists of long constant polypeptide regions of about 55 kDa and a light chain ("L", the orange part) of about 22kDa . Both regions have a constant ("C") and a

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variable ("V") part. The diversity combinations of variable regions results in the ability to bind different antigens and provides a variety targets of immunoglobulin. The H and L chains are bound by disulfide bonds (black links in the figure mentioned).

The higen region, composed of one or two disulfide bonds between the two heavy chains, is responsible for the great flexibility of the immunoglobulin Y-shape.

The most common type of antibody is immunoglobulin G (figure1.8), it composed about 75 % of immunoglobulins present in blood serum.

(A) (B)

FIGURE1.8: Schematic representations of Immunoglobulin

G structure.

The IgG molecule is composed by constant (C) and variable (V) domains for each light (L) or heavy (H) chain: VL, CL, VH, CH.

Its Y-shape is usually decomposed in two regions: two Fragments Antigens-Binding Fab which are the tips that interacts with the antigens and a

Frag-ment crystallizable Fcpart.

IgG is often used as a biomarker of unhealthy situations, e.g. increased level of immunoglobulin G in serum indicate possible infections, allergies, autoimmune disorders, cirrhosis, chronic inflammations, and its microscopic distribution is also investigated in relation to cancer [48].

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9

Chapter 2

Coherent control

The possibility to measure, to unveil and manipulate ultrafast phenomena is an intriguing goal in physics and chemistry. As Werschnik and Gross out-line in their paper [49], even if laser was invented in the 1960’s only in the last decades an application of Quantum Optimal Control Theory (QOCT) was possible. This is mainly due to the advent of femtosecond lasers in the 1980’s and the growing ability to create tailored pulses thank to the techno-logical advances.

The Quantum Optimal Control Theory is based on the capability to op-timize the experimental tools until a desired product is obtained.

Optimal control experiments cover a wide domain of topics, including: con-trol over isomerization of proteins [50], control over molecular dissociation and ionization [51,52] , fragmentation [53,54,55], control of attosecond dy-namics [56], and many other applications.

As an example of quantum control, the selection of chemical reactions for a triatomic molecule can be mentioned (see fig.2.1) [2,3, 57]. The

Tannor-(A) (B) (C) (D)

FIGURE 2.1: Example of control over the evolution of

a wavepacket. A transition from the local minimum in the groundstate of a triatomic molecule ABC (B) to the excited state generates a wavepacket. A second interaction induces the wavepacket to end up in one of the two different products of the groundstate : A+BC (A) or AB+C (C). (D): Sketch of the

Tannor-Kosloff-Rice scheme. Taken from [2,57]

Kosloff-Rice scheme depicted in figure2.1dshows that a wavepacket is cre-ated on the first excited state by an ultrashort pump pulse; with the help of a

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10 Chapter 2. Coherent control

second laser pulse of the proper wavelength and time delay with respect to the excitation event, the wavepacket can be dumped into the desired prod-uct channel. Thus, different reaction channels can be accessed selectively depending on the pump-dump delay time . An experimental application of Tannor-Kosloff-Rice is reported in Baumert’s article studying Na2

prod-ucts: as it is shown in fig.2.2, there is a variation of the ratio between Na+₂ and Na+products by adjusting the pump-probe delay time [58].

FIGURE 2.2: Controlling the Na+2 production versus the

Na+ production in a Tannor-Kosloff-Rice-like scheme by adjusting the pump-probe delay time. Taken from [58].

As part of quantum control, the Coherent Control Theory focuses on the ability to design a field that can drive molecules to the desired final states. In theory, optimally laser pulse can be calculated by solving the time-dependent Schroedinger equation of the system; however a full description of the evo-lution of the wavepacket as well as a priori knowledge of Hamiltonians are needed for this purpose. In particular, for a complex system, such as large molecules in the condensed phase, the molecular Hamiltonian is known usually to a limited degree and solving the Schroedinger equation is chal-lenging.

Most important, even if a field is generated theoretically on the basis of an approximate Hamiltonian, it may not be sufficient suitable due to errors arising from the Hamiltonian itself as well as from uncertainties coming from the laboratory.

However, tailored laser pulses that steer the quantum system from its ini-tial state to a desired final state can be found using the phase-shape control that is widely studied in different contexts: constructive or destructive in-terferences between quantum paths can coherently manage the excitation probability in a two-photon transition [59, 60, 61] and in the attosecond timescale [62], in chemical processes as isomerization [63], chemical bond breaking [64] or can steer the fluorescence of dye molecules [65]. This ap-proach enables also applications based on quantum coherent evolution in a Josephson-junction qubits [66,67] as well as on quantum information pro-cessing [68].

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Chapter 2. Coherent control 11

The great potentials of the Coherent Control theory are here exploited, applying them to the Optimal Dynamics Discrimination (ODD) theory in-troduced by Judson and co-workers in 1992 [4], in order to discriminate between large molecules with the same absorption and emission spectra taking advantages from their different dynamics. The ODD Theory points out the importance of phase control over the wavepacket and introduces the physical quantity of discrimination (for example fluorescence’s depletion). This target observable is the feedback signal in a multiobjective algorithm and it is iteratively optimized improving the laser pulse characteristics, un-til an optimally shaped laser field is found. In this framework, there is no need to have a priori informations of the molecular system.

In section 2.1, the dynamics of a wavepacket in a molecular potential en-ergy is treated qualitatively, underlining the contribution of the anharmonic term of the potential and two different successful examples of discrimi-nation are shown. The ODD theory is presented in section2.2, highlight-ing the importance of phase control. Finally, the multiobjective algorithm NSGA II used in the experiments is explained in section2.3.

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2.1 Coherent control over a molecular wavepacket

In condensed matter, the interaction between a molecule and an electro-magnetic field leads to a new arrangement of the electronic density and the molecule is said to be electronically excited. Quantistically, the light-matter interaction is described by the time dependent Schroedinger equation

ı∂

∂t|Ψ(r, t)i = H |Ψ(r, t)i H = H0− µ (t)

H0= T + V (r)

(2.1)

in the atomic units : ~ = m = e = 1. Here |Ψ(r, t)i is the wave func-tion, µ is the dipole operator, (t) is the time-dependent electric field and T and V are the kinetic and potential energy operators respectively. The latter is a tricky term that makes an analytical solution somewhat challeng-ing, depending on how many atoms compose the molecules, how and how much they interact with each other and in which environment they are (i.e. including terms as anharmonic vibrational couplings, fluctuations, electro-static interaction between solvent and solute,etc.).

As the mass of the atomic nuclei is ∼ 103times heavier than electrons’s mass, their velocities are slower. As a consequence, while the electrons move through the electronic states during a transition, the nuclear coordi-nates of the molecules does not change. This is summed up in the well known Franck- Condon principle, depicted in figure (2.3): the electronic levels are delineated with a morse-like potential that is harmonic for the lowest energetic states; the horizontal lines are the vibrational eigenvalues ωi.

FIGURE2.3: Representation of Franck-Condon principle

If the spectral width of the pump-pulse is broader than the vibrational spac-ing and short in time compared with the vibrational period of the molecule,

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2.1. Coherent control over a molecular wavepacket 13

several vibrational states become populated generating a coherent superpo-sition of wave functions called "wavepacket". It evolves in time through the quantum mechanical phase space as

|Ψ(t)i =X

i

ci e−ı(ωit+φi)|ψii (2.2)

where |ψii are the eigenfunctions, ci and φi represent their amplitudes

and phases respectively and ωi are the transition frequencies of the ith

vi-brational levels.

When the molecular wavepacket is generated, it periodically oscillates back and forth in the bound state potential energy surface. If there are no external perturbations, it continues oscillating without losing energy until it fluoresces decaying from the excited state. Otherwise, it could undergo an-other transition to a second excited state interacting with the probe-pulse. The variation of fluorescence due to the depopulation of the first excited state is called "Depletion". As it is pointed out in the following sections, it is the physical observable for discriminate between molecules using a pump-probe technique.

In the case of asymmetric potential that is usually associated with the introduction of a Morse-like potential that brings an anharmonic term, the wavepacket undergoes a broadening (see fig.2.4).

FIGURE2.4: Broadening due to the potential’s anharmonic

term. Adapted from [69]. In fact, the Morse potential

V (R) = Dee−2α(R−R0)− 2e−α(R−R0)

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where −Deis the energetic minimum calculated in the equilibrium distance

R0, brings to the eigenvalues

Eν = ~ω0(ν + 1 2) − β(ν + 1 2) 2 (2.4) with β = ~ω 2 0 4De

known as the anharmonicity constant.

The anharmonic term brings about a variation of the vibrational energies’s spacing ∆Eν = ~ω0 − 2β~ω0(ν + 1)that is smaller for the highest states

∆EH < ∆EL, leading to different oscillation periods TH > TL. As a

conse-quence, the lower energy ("red") components of the molecular wavepacket advance the higher energy ("blue") components after some oscillations [69,

70,71]. In addition to the wavepacket dispersion, the anharmonicity causes another phenomenon: the decoherence. Thus, the bigger is the anhar-monicity, the shorter is the time until the wavepacket has completely spread out on all the possible configurations and the initial phase’s information is lost.

The manipulation of both the phase φn and the amplitude cn of each

component of laser pulse, namely the "Coherent Control" approach, can suppress the wavepacket dispersion, making possible to collected all the components of the wavepacket in phase at a specific delay time T short enough to ignore the decoherence effects (shorter than a few picoseconds). As a result, there is a significant enhancement of the fluorescence depletion and the example in section2.2.1clarifies the importance of this argument.

2.1.1 Examples of molecular discrimination

GAP Biophotonics group has already succeeded in discriminating between molecules with the same absorption and emission bands and fluorescence’s dynamics. Thank to the use of tailored pulses, they measured a variation of the time-resolved fluorescence [7,8,9].

Figure 2.5 shows a scheme of the pump-probe technique that is used to acquire a time-resolved fluorescence trace. At time t = 0, a short pulse excites molecules from the ground state to the excited state state with a defined phase conditions, forming a coherent superposition of vibrational states. The molecular wavepacket evolves in time differently in the two molecules under investigation and the evolution can be probed at time t = T by a second pulse, which transfers the population to an higher state (ionizing and dissociative), depleting the fluorescence signal. Thus, a time-resolved fluorescence trace can be acquired changing the time T .

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FIGURE 2.5: Pump-probe scheme for acquiring a

time-resolved fluorescence trace.

The first result was achieved in 2009 by the work of Roth et alii apply-ing ODD to discrimination of riboflavin (RBF) and flavin mononucleotide (FMN) [7]. Not only was it possible to distinguish two otherwise spectro-scopically indistinguishable molecules, but retrieval of their relative con-centrations in a mixture was demonstrated, thereby making ODD directly applicable for concrete applications in chemistry and biology.

More recent result involves Rondi’s paper [8]: the two molecules inves-tigated, Tyrosine (Tyr) and Tryptophan (Trp), are two of the 20 amino-acids that made up proteins and the main contributors in protein’s fluorescence.

FIGURE 2.6: Time-resolved Fluorescence depletion for Trp

(black) and Tyr (red). (a) pulse shaped (b) pulse unshaped. From [8]

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They are characterized by the same absorption band centred at 270 nm and fluorescence bands around 350 nm for Trp and around 310 nm for Tyr. Pulse-shaping technique is used here to manipulate the phase of the UV field inducing different time-resolved fluorescence depletion traces (upper panel in fig. 2.6) with a significant signal’s variation of ∼ 35%. In fact, Trp undergoes rapid fluorescence depletion reaching a minimum at 600 fs, which can be attributed to the opening of a Franck-Condon window to-ward higher lying ionizing states. In contrast, Tyr fluorescence decreases until 600 fs and then continues less abruptly until 7 ps. Lower panel in fig-ure2.6 shows that the same time-resolved fluorescence trace is measured with unshaped UV pulse, making impossible a dynamical discrimination.

The second example concerns the discrimination between Tryptophan and Ala-Tryptophan (ala-Trp) [9]. The absorption and emission bands are measured for both molecules and collected in figure2.7a. In this case, the fluorescence emission band of Ala-Trp coming from the chromophore is only barely perturbed by the alanyl residue.

(A)

(B)

FIGURE2.7:

(A): Absorption and emission bands of some compounds. (B): T-R Fluorescence depletion for Trp (blue) and Ala-Trp (red).

(a)pulse shaped (b) pulse unshaped. Taken from [9].

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a different dynamics in the case of UV pulse shaped. This former experi-ment, involving two systems with almost equivalent energetic structures, suggests us that a similar result can be obtained for tryptophan contained in larger molecules as proteins.

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2.2 Optimal quantum control

Taking advantages of the formulation contained in Li’s paper [72], let us consider an example of quantum systems represented by multiple chemi-cal species, that we intend to discriminate. Each of them are characterized by N active control states |φν0i,|φν1i,. . . ,|φνN −1i and a detection level |Γνi .

For each νth species, the solution of the time dependent Schroedinger equa-tion is a wavepacket that is a superposiequa-tion of all possible states

|ψν(t)i =

N −1

X

i=0

cν_i(t) |φν_ii + dν(t) |Γνi (2.5)

Let suppose that at time zero only the ground state is populated

|ψν(0)i = |φν₀i (2.6)

After the first interaction c(t) that lasts 0 6 t < T , only the |φνii with

0 < i < N − 1are occupied |ψν_{(T )i = U}ν c(T, 0) |ψν(0)i = N −1 X i=0 cν_i(T ) |φνi (2.7)

where Ucν(T, 0)is the propagator describing the evolution of the initial state

under the influence of the field c(t). As the detection state is not populated

dν(T ) = 0, thus the normalization constrain is

N −1

X

i=0

|cν_i(T )|2= 1 (2.8)

At time T the second interaction d(t)comes and interacts for T 6 t 6 T0;

thereby the wavepacket becomes |ψν_{(T + T}0_{)i = U}ν

d(T

0_{, T ) |ψ}ν_{(T )i} _(2.9)

where Udν(T0, T )is the propagator from time T to T0.

Let us suppose that our system is controllable in the sense explained by Werschnik et alii in [49], our goal is reduced to find a measurable quantity Jν, that could change among the species, and a tailored laser pulse that can maximise it only for a ξth species :

Jξ= max

(t) J

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2.2. Optimal quantum control 19

where J that is defined as the expectation value of an operator Oν. The latter is usually a projector over a single state, in our case the state of our system after the two interactions (at time T + T0)

Oν = |ψiν(T + T0)i hψiν(T + T0)| (2.11)

but we can also employ a multi-objective target operator defining Oν =X

i

βiOiν (2.12)

Oν could also be define in other ways: as local or non-local operator de-pending on the specific system taken into account [49]. The sole restriction is that Oνhas to be an hermitian operator.

If the operator taken into account is the one described in eq.(2.11) and the transition from the ground state |φν0i to the final one |Γνi is negligible, as

it is in our experimental condition described in section 4.1, our signal is a depletion of fluorescence from the excited states |φνii with i 6= 0.

Using equations (2.7) and (2.9) Jν = hΓν|ψν i(T + T0)i hψiν(T + T0)|Γνi = | N −1 X i=0 cν_i(T ) hΓν| U_dν|φν_ii |2 (2.13)

If energetical structure does not change too much among the species hΓν| U_dν|φν_ii ≡ Dν_i ≈ Di ∈ R, thus Jν = | N −1 X i=0 cν_i(T )Di|2= |cν(T ) · D|2 (2.14)

If we want to detect the ξth species among a mixture of different ν com-pounds one of the possible relevant quantity that we investigate is1

L= Jξ−X

ν6=ξ

Jν (2.15)

Finally, L is maximized when we find the shape of the laser pulse c(t)

which drives the transition from an initial state of the νth species |φν₀i to a final state |ψiν(T )iin such a way that Jξis optimized and all the other Jν

for ν 6= ξ are minimized . This happens if the scalar product in eq.(2.14) is maximum: when the vector cξ(T ) is parallel to D. Otherwise, Jν are

1_{We could add several other terms in order to get closer to an optimized description of}

the system: requiring for example that the power losses due to the dispersion in the perpen-dicular direction of the laser field is minimal, or that the final state satisfies the Schroedinger equation at each time [49]. Even if these are proper conditions they are not necessary in this case.

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minimized if the scalar product is zero: here cν(T )is perpendicular to D (see fig.2.8).

FIGURE2.8: Representation of the conditions for

discrimi-nation between species. Adapted from [72].

In the next paragraph we made a very simple example that clarifies the condition written above, outlining the importance of the phase control over the wavepacket.

2.2.1 Example: three levels system

Let us consider a 3 levels system composed of a groundstate |0i_ν , an ex-cited state |1i_ν and a detection level |Γνi . The first pulse cdrives the

tran-sition from |0i_ν to |1i_ν while the pulse dfrom |0iν to |Γ

ν_{i and from |1i}

ν to

|Γνi (see figure2.9).

FIGURE2.9: Three level system scheme:

at time zero cinteracts with the ground state populating the first

excited state, while at time T the second pulse leads the transitions to the detection level.

Let us suppose that there are different probabilities to enhance the second transitions: for example D0 = 0and D1 = D. Thus, referring to eq.(2.14)

Jν = | 1 X i=0 cν_i(T )Di|2 = |cν₀(T )D0+ cν1(T )D1|2 = |cν₁(T )D|2 (2.16)

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2.2. Optimal quantum control 21

As a consequence, the more we populate the state with the highest proba-bility of a second transition (|1i_ν in this case) the more the signal increases. In fact a parallel vector of the form cν_// = (0, cν₁(T ))has a greater feedback instead of a perpendicular one cν⊥= (cν0(T ), 0). Moreover, as D doesn’t

de-pend on the specific species due to the fact that the energetic structures are almost the same for the species taken into account, the signal depends on the value of the coefficient of the state that will be depopulated only at time T, when the second transition occurs.

A second argument is the following: from eq.(2.7) the evolution in time of the wavepacket is

|Ψν_{(T )i = c}ν

0(T ) |0iν+ cν1(T ) |1iν (2.17)

So, from eq.(2.14) reminding that cν_i(T )is a complex number c_iν(T ) = cν_ieıφνi(T )

while Di∈ R Jν = Re(cν₀(T )D0+ cν1(T )D1 2 + Im(cν₀(T )D0+ cν1(T )D1 2 = (cν₀D0)2+ (cν1D1)2+ 2(cν0D0)(cν1D1) cos(φν0(T ) − φν1(T )) (2.18)

Therefore, it is evident that the optimization of the signal relies also on a specific phase’s difference condition between the two states at time T . In particular max φi Jν implies cos(φν₀(T ) − φν₁(T )) = 1 leading to φν₀(T ) = φν₁(T )

Thus, the wavepacket’s components should be in phase at time T (see fig.

2.5) to enhanced the depopulation of the excited state.

This case could be easily extended to the general one with N+1 states

Jν = Re( N X i cν_i(T )Di) 2 + Im( N X i cν_i(T )Di) 2 = N X i (cν_iDi)2+ 2 N X i X j6=i

cν_iDicνjDj[cos(φνi(T )) cos(φνj(T )) + sin(φνi(T )) sin(φνj(T ))]

= N X i (cν_iDi)2+ 2 N X i X j6=i cν_iDicνjDjcos(φνi(T ) − φνj(T )) (2.19) that leads to φνi(T ) = φνj(T )for i 6= j. This simple example outlines the

great importance of phase control over the wavepackets to discriminate be-tween differente species.

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2.3 Multiobjective Genetic Algorithm

After Judson and Rabitz’s work [4], closed loop learning procedures based on genetic algorithms have contributed to achieve coherent-control’s exper-iments.

An example of the closed-loop learning algorithm scheme is shown in fig (2.10) : at first step the Genetic Algorithm (GA) starts generating an ini-tial random population of pulse’s shapes, that is materialized by a pulse shaper and applied to the molecular system. The observable, for example fluorescence, is recorded by a suitable detector and then compared. Step by step, an optimally shape should be found by converging to the best solution found by the GA.

FIGURE2.10: Closed-loop learning algorithm scheme.

Genetic algorithms are optimization methods based on several metaphors from biological evolution. Firstly, let consider a population of N "parents solutions" Pi of the ith step of the closed-loop method. With a mixture of

this possible pulse shapes, we can collect a group of N "children solutions" Qithat acquires a combination of parents genes.

Secondly, the population of children solutions can be affected by some mu-tations that leads to a corruption of genes, with a low rate of probability. Finally, as the solutions Pi and Qi are not equally suitable, half of the

pop-ulation is rejected. As a result, only a few individuals among the global population can pass their genetic information onto succeeding generations (the i+1th step).

The approach called Elitist Non-Dominated Sorted Genetic Algorithm (NSGA-II), developed by Deb et al. [6], is capable to handle simultaneously several targets and increases the versatility and the robustness of the GA approach [5].

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2.3. Multiobjective Genetic Algorithm 23

In the Optimal Dynamics Discrimination point of view, the optimiza-tion goal is the selective enhancement of the signal from a specific molecule in competition with a second one, characterized by similar or overlapping absorption and emission bands. Thus, one of the target objective is typi-cally defined as the ratio between the signals simultaneously generated by the two systems

L₁ = J

ξ

Jν (2.20)

the second target is the signal we want to maximize or minimize for exam-ple

L₂= Jξ (2.21)

The other features that increase the potentialities of NSGA-II approach com-pared with GA are the following points, depicted in figure2.11.

Nondominated sorting approach.

The parents and children solutions Pi and Qi are sorted according to

non-dominant definition: a solution dominates another solution, if it is not worse in any of the objectives, and is strictly better in at least one. A so-lution is called nondominated, if there is no soso-lution that dominates it. If some equivalent solutions are nondominated, they are ranked in the first front F_i1 of the ith step. If the size of F_i1is smaller than N, all its members are retained in the next parent population Pi+1. The remaining individuals

of Pi+1are selected in the next nondominated front Fi2, which is the group

of solutions that are dominated only by F_i1 individuals. The process ends when the parent population Pi+1is filled .

FIGURE2.11: Scheme of NSGA II procedure.

The individuals are sorted according to nondominant definition into non-dominated fronts. When the new parent population is filled, the solutions are submitted to the crowding-distance

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Diversity preservation.

After the parent population Pi+1is filled, the individuals are sorted

accord-ing to the crowdaccord-ing distance operator, which is introduced for preservaccord-ing the population diversity.

The crowding distance is an estimate of the density of solutions: it is the perimeter of cuboid formed using the nearest neighbors of the ith solution as the vertices and it is calculated by NSGA-II (see figure2.12). Diversity is preserved by retaining the solutions that have the bigger crowding dis-tances for the next generation.

FIGURE 2.12: Crowding distance calculation for the

solu-tions (filled circles) belonging to the same pareto front (or nondominated front) using two objectives J1and J2. From

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25

Chapter 3

Transient Absorption and

Stimulated Raman

Spectroscopies

Transient absorption technique was developed more than 60 years ago by Manfred Eigen, Ronald G. W. Norrish and George Porter, awarded in 1967 for this important achievement with the Nobel Prize in Chemistry [73]. They firstly developed experimental apparatus with millisecond time reso-lution based on extremely powerful arc flash lamps as excitation sources.

The stirring technological achievements of recent years, particularly the advent of femtosecond laser sources paved the way to monitor photophys-ical and photochemphotophys-ical processes, either intramolecular or intermolecular, that follow the absorption of a photon by a molecule on picosecond and subpicosecond timescales. When a ground state molecule absorbs a pho-ton it is promoted to an excited state. From this higher energy state the molecule can relax back to the ground state via some other intermediate excited state of singlet or triplet spin multiplicity [74] or chemically react to form products i.e. photodissociations and photoionizations may occur [75,76]. The formation of polaron pairs, polarons, and triplet excitons [77], charge recombination [78], electron-phonon relaxation [79] as well as other forms of cooling [80] can be monitored by detecting the absorption spec-trum in suitable wavelength windows, typically in the UV-Vis-NIR range. Photoisomers and other metastable species can also be investigated [81] .

Although the transient absorption technique gives informations about the electronic reconfiguration that follows the interaction with the pump pulse, it is lack of structural sensitivity: it may lead to contrasting scenar-ios regarding the change in the molecule’s reconfiguration as testified for example by the different hypotheses formulated on hemeproteins photo-dynamics and based on TA measurements [82, 83]. As a result, a kinetic analysis cannot be achieved unambiguously by a single spectroscopic tech-nique.

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26 Chapter 3. Transient Absorption and Stimulated Raman Spectroscopies

It is essential to separate the contributions from the often simultaneous events that may occur. For this purpose, Femtosecond stimulated Raman spectroscopy (FSRS) is a powerful tool: providing high spectral resolution and high time precision in triggering the generation of vibrational coher-ences (∼ 50 fs), it enables the measurement of specific molecular vibrational structure [11]. In addition, under electronic resonance condition, the FSRS technique is able to selectively enhance different cromophores or electronic states, removing the ambiguities created by the TA measurement [84,81].

Femtoscopy group succeded in combining the two techniques to unveil the photodynamics of methyl-phenylthiophene (MPT), one of the build-ing block of photoactive materials [85]. The transient absorption of MPT (figure3.1a) shows the evolution of the absorption outlining the formation of two contributions: the first one at early times (∼ 500 fs) centred at 480 nm and the second one around 366 nm which appears later (∼ 100 ps). The FSRS measurements (figure 3.1b) unveils the dynamical evolution of this two transient species, selectively isolating the contributions tuning the Raman pulse (RP) into electronic resonance with the singlet (480 nm) and triplet (366 nm) excited states.

Moreover, the article is an example of the importance of resonance con-dition: the FSRS spectra at both RP wavelengths show a strong resonant enhancement with respect to the (off resonance) ground-state stimulated Raman (SR) spectrum measured at λRP = 366 nm. In this case the FSRS

suits better than SR technique, increasing the signal from ∼ 0.1% to ∼ 1%.

In this chapter we introduce the density matrix formalism and the inter-action picture (section3.1) that are useful to deal with nonlinear polariza-tion. The χ(3)processes are described with diagram theory (section3.1.1) in order to outline the most important contributions to the transient absorp-tion signal (secabsorp-tion3.2) . Finally, in section3.3 we introduce the femtosec-ond stimulated Raman scattering technique.

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Chapter 3. Transient Absorption and Stimulated Raman Spectroscopies 27

(A)

(B)

FIGURE 3.1: Example of photodynamical study achieved

with a combination of TA (A) and FSRS (B) techniques. Taken from [85].

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3.1 Interaction picture and diagram theory

The solution of the Schroedinger equation d

dt|ψ(t)i = − ı

~H |ψ(t)i (3.1)

is a wavefunction that can be expressed as a coherent superposition of eigenstates |ni with complex coefficients cn

|ψi =X

n

cn|ni (3.2)

we can define the probability to find the system in the eigenstate |ii

Pi= | < i|ψ > |2= |ci|2 (3.3)

Similarly, we can define the density matrix ρ := |ψi hψ| =X

n,m

ρn,m|ni hm| (3.4)

where ρn,m = cnc∗m. This formalism allows us to define the population

states (n = m) and the coherences between pure states (n 6= m). Moreover, defining

ρ :=X

k

Pk|ψki hψk| (3.5)

where Pkis the probability, the density matrix permits to treat the statistical

ensembles [86].

The Schroedinger equation in the density matrix formalism leads to the Liouville-Von Neumann equation

d dtρ(t) = d dt(|ψi hψ|) = − ı ~ H, ρ(t) (3.6)

Moreover, in quantum mechanics the expectation value of an operator A is defined as

hAi_def = hψ| A |ψi =X

n,m

cnc∗mAm,n (3.7)

while it becomes in the density matrix’s formalism hAi_def =X

n,m

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3.1. Interaction picture and diagram theory 29

The interaction picture

If the Hamiltonian is time-dependent trough an interaction term H0(t) which is weaker compared to time-independent part H0, the system’s

evo-lution can be treated perturbatively, in the so-called "interaction picture". The Hamiltonian related to a light-matter interaction can be described in the semiclassical limit as

H(t) = H0+ H0(t) = H0+ µE(t) (3.9)

where µ is the quantic dipole operator and E(t) is the classical electric field. Let us define the wavefunction in the interaction picture as

|ψ(t)i = U₀(t, t0) |ψI(t)i (3.10)

here U0(t, t0) = e− ı

~H0(t−t0)is the propagator related to the Hamiltonian H₀.

HI(t) = U0†(t, t0)H0(t)U0(t, t0) (3.12)

Solving the Schroedinger equation for the interaction picture (equation3.11) iteratively in the density matrix formalism (eq.3.6) leads to

ρI(t) = ρI(t0) + − ı ~ Z t t0 dτHI(τ ), ρI(τ ) = ρI(t0) + − ı ~ Z t t0 dτHI(τ ), ρI(t0)+ + − ı ~ 2Z t t0 dτ Z τ t0 dτ1HI(τ ),HI(τ1), ρI(τ1) = ρI(t0) + ∞ X n=1 − ı ~ nZ t t0 dτn. . . Z τ2 t0 dτ1HI(τn), . . .HI(τ1), ρI(t0) . . . (3.13)

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Thereby, the nthorder component of the density matrix with respect to HI

interaction is ρ(t)(n)= −ı ~ nZ t t0 dτn. . . Z τ2 t0 dτ1U0(t, t0)HI(τn), . . .HI(τ1), ρ(t0) . . . U † 0(t, t0) (3.14) where HI(τk) = U0†(τk, t0)µE(t)U0(τk, t0) = E(τk)U0†(τk, t0)µU0(τk, t0) = E(τk)µI(τk) (3.15)

While in the Schroedinger picture the dipole operator is time independent, in the interaction picture it depends on the propagator U0(τk, t0).

The nonlinear polarization

In the case of linear optics the induce polarization depends linearly on the electric field, accordingly to

P = 0χ(1)E

where 0is the permittivity of free space and χ(1)is the linear susceptibility

tensor. The nonlinearities, that can be exploited by the great amount of power released by pulsed lasers, are described in terms of a power series in the field strength. The power of E or the χ’s superscript reveal the order of nonlinearity .

P = 0

χ(1)E + χ(2)E2+ χ(3)E3+ . . .

In χ(n)processes there are n interactions among n different fields or it may happen that the same field interacts more than once with the system.

The macroscopic polarization is given by the expectation value of the dipole operator

P (t) := hµ(t)i_def → P (t) = Tr(µ(t)ρ(t)) (3.16) Thus, assuming that ρ(t0) is an equilibrium density matrix that does not

evolve in time we can send t0 → −∞.

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the nthorder nonlinear polarization

P(n)(t) = − ı ~ nZ t −∞ dτn Z τn −∞ dτn−1. . . Z τ2 −∞

dτ1E(τn)E(τn−1) . . . E(τ1)·

hµ_I(t) ·µI(τn),µI(τn−1), . . .µI(τ1), ρ(−∞) . . . i

(3.17) Changing variables from instant variables to time intervals τ1 = 0; τ2 −

τ1 = t1; . . . and forgetting about the subscript I of the dipole operator, the

equation becomes P(n)(t) = − ı ~ nZ ∞ 0 dtn Z ∞ 0 dtn−1. . . Z ∞ 0 dt1

E(t − tn)E(t − tn− tn−1) . . . E(t − tn− tn−1− · · · − t1)·

hµ(t_n+ tn−1+ · · · + t1) ·µ(tn−1+ · · · + t1), . . .µ(0), ρ(−∞) . . . i (3.18) or equivalently P(n)(t) = Z ∞ 0 dtn Z ∞ 0 dtn−1. . . Z ∞ 0 dt1

E(t − tn)E(t − tn− tn−1) . . . E(t − tn− tn−1− · · · − t1) · Sn(tn, tn−1, . . . , t1)

(3.19) where S(n)({tn}) is the nonlinear response function

Sn({tn}) = −ı ~ n hµ(tn+ · · · + t1) ·µ(tn−1+ · · · + t1), . . .µ(0), ρ(−∞) . . . i (3.20)

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3.1.1 Double sided Feynman diagrams of a χ3 process

In the case of a third order nonlinear process, the polarization is described by the product of three fields

P(3)(t) = Z ∞ 0 dt3 Z ∞ 0 dt2 Z ∞ 0

dt1E(t−t3)E(t−t3−t2)E(t−t3−t2−t1)S(3)(t3, t2, t1)

(3.21) while the response function is the expectation value of the product of four dipole operators that leads to a Four-Wave Mixing (FWM): three dipole are introduced by the Hamiltonian through the interaction with the elec-tric fields, the forth, that is out of the commutators, comes from the free induction decay. S(3)(t3, t2, t1) = −ı ~ 3 hµ(t3+ t2+ t1) ·µ(t2+ t1),µ(t1),µ(0), ρ(−∞)i (3.22) Thereby, unrolling the previous equation we note that the commutators make the dipoles act on both sides of the density matrix. The signal ac-quired is composed of the sum of all the following contributions that are depicted in fig3.2. + hµ(t3+ t2+ t1) · µ(t2+ t1)µ(t1)µ(0)ρ(−∞)i ⇒ S3 − hµ(t₃+ t2+ t1) · µ(t1)µ(0)ρ(−∞)µ(t2+ t1)i → S4∗ − hµ(t3+ t2+ t1) · µ(t2+ t1)µ(t1)ρ(−∞)µ(0)i → S2∗ + hµ(t3+ t2+ t1) · µ(t1)ρ(−∞)µ(0)µ(t2+ t1)i → S1∗ − hµ(t3+ t2+ t1) · µ(t2+ t1)µ(0)ρ(−∞)µ(t1)i ⇒ S1 + hµ(t3+ t2+ t1) · µ(0)ρ(−∞)µ(t1)µ(t2+ t1)i ⇒ S2 + hµ(t3+ t2+ t1) · µ(t2+ t1)ρ(−∞)µ(0)µ(t1)i ⇒ S4 − hµ(t₃+ t2+ t1) · ρ(−∞)µ(0)µ(t1)µ(t2+ t1)i → S3∗ (3.23)

Here, Si∗ are the complex conjugate of Si because of the trace invariance

under cyclic permutation: e.g. Tr(ABC) = Tr(BCA)=Tr(CBA) .

The Feynman’s diagrams, reported in figure3.2, are a common way to visualize the nonlinear response function and they can be interpreted by means of the following rules.

1. The last interaction must end in a population state of the form |ni hn|. 2. Vertical lines represent the time evolution of the ket and bra of the density matrix, while the time is running from the bottom to the top. The initial ground state is ρ(−∞) = |Ai hA|

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3. Interactions with the light field are represented by arrows. An arrow pointing towards the system represents an up-climbing of the corre-sponding side of the density matrix due to the interaction with the field E0e−ıωt, while an arrow pointing away represents a de-excitation

due to interaction with the pulse E0eıωt.

4. The last interaction represents the signal generated by free induction decay; it is indicated with a dashed line and must points away (be-cause of rule 1). The emitted light has a frequency and wavevector which is the sum of the input frequencies and wavevectors (consider-ing the appropriate signs).

5. Each diagram has a sign (−1)m, where m is the number of interactions from the right, because of each time an interaction is from the right in the commutator it carries a minus sign.

FIGURE 3.2: Feynman’s diagrams related to a χ3 process

composed by the contributions outline in eq.3.23.

All these terms illustrated above and in figure 3.2 are obtained under the rotating wave approximation (RWA): we are neglecting all the response functions that do not conserve the energy as the diagrams with de-excitation from the ground state, etc. In particular, that approximation removes the terms with high frequency oscillations that are neglectable in resonant con-dition.

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3.2 Ultrafast Transient Absorption Spectroscopy

In transient absorption (TA) spectroscopy, a fraction of the molecules is pro-moted to an electronically excited state by means of an excitation (or pump) pulse. Depending on the type of experiment, this fraction typically ranges from 0.1% to 10 %. Figure3.3illustrates TA technique: a probe pulse whose intensity is low enough to avoid multiphoton/multistep processes during probing, is sent through the sample with a temporal delay τ with respect to the pump pulse. After the sample, the pump beam is blocked while the probe spectrum is acquired with and without the presence of the pump.

FIGURE3.3: Transient absorption experimental scheme.

Adapted from [87].

Measuring the absorption from both photoexcited AON(τ, ω)and un-photoexcited AOF F(ω)molecules’ spectra enables to define the differential transient ab-sorption as ∆A(τ, ω) = − log A ON_{(τ, ω)} AOF F_(ω) (3.24) Defining AON(τ, ω) = |EON(τ, ω)|2and AOF F(ω) = |EOF F(ω)|2we get

∆A(τ, ω) = − log _E S+ E_{N L}(3) 2 ES 2 = − log 1 − 2Im(ESP (3)₎ |E_S|2 ∝ 2Im(ESP (3)₎ |ES|2 ∝ Im(P(3)) (3.25) where we set E_{N L}(3) = −ıP(3) (3.26)

By changing the time delay τ between the pump and the probe and recording a ∆A spectrum at each time delay, a ∆A(τ, ω) map as a function of τ and frequency ω is obtained.

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3.2. Ultrafast Transient Absorption Spectroscopy 35

In general, a ∆A spectrum contains contributions from various 3rd order nonlinear processes that are outlined in figure3.4. They are here explained referring to the diagram theory reported in section3.1.1and in particular to figure3.23.

FIGURE3.4: A typical TA spectrum (solid blue line) and its

components: ESA (solid line), SE (dotted line), GSB (dashed line).Adapted from [87]

The diagrams are built under the constrains: 1. the pump pulse interacts first.

2. the pump pulse interacts twice simultaneously.

As a consequence

3. the probe is the last interaction delayed with respect to pump field τ = t2.

4. the free induction decay’s frequency is equal to ωPROBE, thus

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1. Excited-State Absorption (ESA), diagram S1.

The population of the excited state |Bi hB| is due to two simultaneous interactions with the pump. Optically allowed transitions from this state to higher excited states may exist in certain energy regions, and interaction with the probe pulse at the corresponding wavelengths will occur leading to a coherent state |Ci hB|.

A free induction decay leaves the system to the population state |Bi hB| within the coherence time t3<

1 ΓCB

, where Γ is the dephasing rate. The response function associated to this process is

S(3)_ESA= − ı ~3

hµ(t₃+ t2) · µ(t2)µ(0)ρ(−∞)µ(0)i (3.27)

and with monochromatic fields leads to ∆A(ω)_ESA∝ µ 2 ABµ2BC|Epump|2Eprobe ~3 δ(ω − ωprobe)ΓCB ((ωCB− ωprobe)2+ Γ2CB) (3.28)

Thus a positive signal in the ∆A spectrum is observed in the wave-length region of excited-state absorption represented as a solid black line in figure3.4.

FIGURE 3.5: Ladder diagram for Excited state absorption

processes.

solid arrow: interaction with the ket; dashed arrow: interaction with the bra; waivy arrow: free induction decay.

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2. Stimulated Emission (SE), diagram S2.

During the physical process of stimulated emission, a photon from the probe pulse induces emission of another photon from the excited state |Bi hC|, which returns to the lower energy state |Ci hC|.

The photon produced by stimulated emission has the same frequency and is emitted in the same direction as the probe photon, and hence both will be detected.

The response function associated to this process is S(3)_SE = ı

~3

hµ(t3+ t2) · µ(0)ρ(−∞)µ(0)µ(t2)i (3.29)

and with monochromatic fields leads to ∆A(ω)_SE∝ −µ 2 ACµ2CB|Epump|2Eprobe ~3 δ(ω − ωprobe)ΓCB ((ωCB− ωprobe)2+ Γ2CB) (3.30) Stimulated emission results in an increase of light intensity on the detector, corresponding to a negative ∆A signal, as schematically in-dicated by a dotted line in fig.3.4.

FIGURE 3.6: Ladder diagram for stimulated emission

pro-cesses.

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3. Ground-State Bleach signal (GSB), diagrams S3and S4.

As a fraction of the molecules has been promoted to the excited state through the action of the pump pulse, the number of molecules in the ground state has been decreased. Thereby, the ground-state absorp-tion in the excited sample is less than that in the non-excited sample. The response function associated to this process is

S_GSB(3) = ı

~3 hµ(t3+ t2) · µ(t2)µ(0)µ(0)ρ(−∞)i + hµ(t3+ t2) · µ(t2)ρ(−∞)µ(0)µ(0)i

(3.31)

and with monochromatic fields leads to ∆A(ω)_GSB∝ −2µ 2 ACµ2AB|Epump|2Eprobe ~3 δ(ω − ωprobe)ΓAB ((ωAB− ωprobe)2+ Γ2_AB) (3.32) Consequently, a negative signal in the ∆A spectrum is observed in the wavelength region of ground state absorption, as schematically indicated by a dashed line in fig. 3.4.

(A)

(B)

FIGURE 3.7: Ladder diagram for ground state bleaching

processes.

(A): S3, (B): S4.

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3.2.1 Singular value decomposition

The singular value decomposition is an useful tool that allows us to extract the spectral components resulting from the different species that occur in the generation of the transient absorption’s signal.

With reference to G.H. Golub arguments [88,89], let us define the decom-position process.

Theorem 3.2.1. If A is a real m-by-n matrix, then there exist orthogonal matrices

U = [u1, . . . , um] ∈ Rmxm and V = [v1, . . . , vn] ∈ Rnxn (3.33)

such that

UTAV = Σ = diag(σ1, . . . , σp) ∈ Rmxn, p = min{m, n} (3.34)

where σ1> σ2> · · · > σp > 0

The σiare the singular values of A, the uiare the left singular vectors of A,

the viare right singular vectors of A and the equation

A = U ΣVT (3.35)

is called the Singular Value Decomposition (SVD) of matrix A.

Corollary 3.2.1. If A ∈ Rmxnand rank(A)=r, then

A = r X i=1 σiuivTi (3.36) Thereby, if A = Ψ(λ, t) Ψ(λ, t) =X n un(t)σnvn(λ) (3.37)

where vn(λ)are the spectral components whose dynamics are controlled by

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3.3 Stimulated Raman Scattering

The Stimulated Raman Scattering (SRS) is a 3rd order pump-probe tech-nique. Thereby, its signal can be obtained as the sum of the processes de-picted in the diagrams reported in section3.1.1 and opportunely dressed with the combination of the fields.

As transient absorption technique, it involves two interactions with the pump pulse and one with the probe field but, on the contrary, SRS does not force the first two interactions to be driven by the pump field. In fact, in a typical SRS experiment, the Raman pulse, which provides the spectral resolution, is temporally overlapped with the the probe pulse and typically lasts a few picoseconds.

The diagrams in figure 3.2 are dressed with each combination of the probe pulse ES(Stokes) and the pump ER(Raman).

Let us consider A the ground state, B the vibrational excited state of the electronic ground state and C the electronic excited state.

1. Stimulated Raman Scattering (SRS) I

SRS I set of diagrams is characterized by having the Stokes field leav-ing the bra side, thus endleav-ing on the population state |Bi hB|. The interactions take place among the energy levels eA6 eB < eC.

FIGURE3.8: Stimulated Raman Scattering I Feynman

dia-grams