Deciphering the non resonant response in impulsive raman spectroscopy

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Sapienza – Universit`

a di Roma

FACOLT `

A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea in Fisica

Tesi di laurea magistrale

Deciphering the non resonant response

in Impulsive Raman Spectroscopy

Candidato:

Lorenzo Monacelli

Matricola 1478892

Relatore:

Prof. Tullio Scopigno

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Introduction

Ultrafast spectroscopy aims to study non equilibrium atomic dynamics at the molecular or condensed matter level, on the femtosecond timescale. This is experimentally achieved using the so called pump-probe approach: a pump beam (for example electrons, X-rays, or optical photons as in the case of the present thesis) prepares the sample in a non-equilibrium configuration, while the probe reveals the state of the evolving system after different time delays.

Using this protocol, several kinds of ultrafast processes can be investigated [1]. Among them, the the retinal’s isomerization involved in the mechanism of vi-sion [2], the ligand dissociation in hemeproteins [3], the opto-magnetic transition induced in Heisenberg antiferromagnets [4, 5].

The field of ultrafast spectroscopy has grown during last two decades thanks to the development of temporal compression techniques capable to synthesise a few fs optical pulses reaching Fourier transform limited bandwidth. This is a key issue for several modern spectroscopic techniques, such as Impulsive Vibra-tional Spectroscopy (IVS) [6]. IVS uses an ultrashort optical pulse to generate vibrational coherence, i.e. impulsive excitation of molecular eigenmodes or op-tical phonons, which is subsequently probed by a second ultrashort pulse at a delayed time.

IVS can measure the excited state vibrational coherence, suppressing off-resonant background signals which commonly affect other kinds of ultrafast experiments [6]. It has been demonstrated in the prototypical cases of photoin-duced dynamics in Bacteriorhodopsin [7], intermolecular vibrational motion in CS2liquid [8], excited state structure of the β-carotene [6] or the excited-state

proton transfer of green fluorescent protein [9]

While a semiclassical theoretical framework correctly describes the frequency of the signal registered with this time domain technique, it fails in predicting the phase dependence of the signal. Critically, this implies that spectral information can only be retrieved based on the modulus of the Fourier transformed, with no access to the rich information contained in the complex phase.

Also, this description applies when the pump pulse induces a vibrational coherence interacting with an electronic state, i.e. in resonance condition. Im-portantly, it can not describe the most common off resonant scenario, when no electronic state participate in the process. This is precisely the focus of the present thesis, which develops a full second quantization theoretical desciption of the IVS, aimed to describe off-resonant signals. To this purpose, we build

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on the diagrammatic perturbative expansion firstly introduced by Mukamel et al ([10, 11, 12, 13, 14]), which allows achieving a physical insight of the non-linear phenomena that generate the signal, providing the key to elucidate the role of the phase. Most important, it allows deciphering the non linear response as function of experimental parameters such as the spectral shape of the probe pulse, its group velocity dispersion [15], and the scattering geometry (the so called phase-matching).

The thesis is structured as follows:

• Impulsive vibrational spectroscopy and the other modern ultrafast tech-niques are presented in chapter 2.

• The background framework of second quantization diagrammatic expan-sion in the Liouville space is presented in chapter 3, and the main ultrafast experimental techniques are fully theoretically described.

• The original theory of IVS is proposed in chapter 4 , discussing the dif-ference between resonant and non resonant technique and describing the signal generated from an unchirped probe. Then the effects of the phase-matching condition, the non collinearity of the beams, the dispersion of the sample and the chirp on the probe pulse are investigated. This chap-ter contains the results obtained by numerical simulations on a simple molecule model.

• The experimental setup to perform IVS experiments is presented in chap-ter 5, together with the experimental benchmarks of the IVS model intro-duced in chapter 4.

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Chapter 2

Ultrafast spectroscopy

In this chapter the field of ultrafast spectroscopy is presented and the main pump-probe experiments are disucssed, fucusing on impulsive vibrational spec-troscopy.

2.1 Transient Absorption

A typical ultrafast optical experiment consists to excite the sample with a fem-tosecond laser pulse, and then to combine different laser beams that access to explore its fast electronic and vibrational out-of-equilibrium dynamic.

This kind of experiments are commonly known as pump-probe. One of the first techniques developed in this way is the Transient Absorption (TA [16, 17]). The Transient absorption consists in measuring the linear absorption of the sample with a probe pulse after an optical excitation (the actinic pulse); tuning the delay between the two pulses.

The signal, measured as the difference between static absorption and tran-sient absorption (without and with the excitation pulse), is a function of the time delay of the two beams, and it collects information about the electronic state dynamics. The population in different electronic states changes when the first laser pulse passes through the sample, and the signal features strictly de-pend on the morphology in the phase-space of the electronic excited states. It is possible to explore the intermediate states during the evolution from reactant to photoproduct.

Figure 2.1 shows a typical TA experiment, performed on the retinal group of the rhodopsin [2].

Critically, the TA alone does not provide information about the vibrational structure of the system. For this reason more sophisticated techniques have been developed, like femtosecond stimulated raman spectroscopy (FSRS, [4, 3, 18]) or impulsive vibrational spectroscopy (IVS, [6]).

The hemeproteins have been studied with most optical ultrafast experi-ments and they represent a good experimental benchmark to compare these techniques [19].

In Figure 2.2 the transient absorption signal of the ligated carboxymyoglobin is reported.

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Figure 2.1: A schematic representation of the transient absorption experiment on the retinal. The actinic pulse excites the 11-cis retinal that is promoted in the electronic excited state S1. Here a fast dynamic takes place, the wave packet drifts through

the isomerization coordinate, and then falls again in the electronic ground state S0

but in a different metastable configuration called all-trans. The probe pulse resonance condition changes in time, if the actinic is sufficiently short in time, the wavepacket generated on the excited state is localized. The central frequency of the stimulated emission generated by the probe in the first part of the dynamic changes as a function of the time delay between pump and probe. After the spontaneous decay of the wavepacket in the all-trans, the probe spectrum is absorbed by the new configuration, so the signal on the probe spectrum changes its sign after the spontaneous decay.

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2.1. TRANSIENT ABSORPTION 9

Figure 2.2: Transient absorption experiment performed on the ligated MbCO hemoglobin. This signal is generated by the protein after the ligand dissociation, with a timescale of 300 fs. Adapted from [20].

(a) (b) TA

Figure 2.3: Hemeprotein transient absorption. The hemeprotein structure on the left, the transient absorption at different times on the right.

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2.2 Ultrafast vibrational experiments

Vibrational spectra can be measured both using infrared photon absorption or Raman scattering [21, 22]. The two techniques are complementary, since the vibrational modes that are Raman-active cannot be usually seen in IR-techniques and vice versa. However, infrared laser pulses are more difficult to be compressed, due the higher transform limit duration, and are absorbed by most used solvents and samples. For these reasons in this thesis we focus only on the Raman technique.

Vibrational dynamics contain information on the molecule structure, geom-etry and on the chemical bounds.

In order to get vibrational information during ultrafast dynamics the Raman and ultrafast spectroscopy have been combined.

2.2.1 Impulsive Spontaneous Raman

Spontaneous Raman provides a very well known way to investigate vibrational frequencies of the system’s ground-state. The use of a pulsed probe after a pump pulse (which excites the sample) enables to get a time resolved Raman spectrum, able to investigate picosecond vibrational dynamics. However the spectral and temporal resolution for transient Raman experiments are limited by the Heisenberg principle:

∆ω∆T ≥ 1 2

This leads to a spectral resolution up to 20 cm−1 _{for pulses at least 500 fs}

long. So sub-picosecond vibrational dynamics cannot be investigated directly using spontaneous Raman without facing a bad spectral resolution.

This limit prevents the studying of sub-picosecond vibrational dynamics, as shown in Figure 2.4.

2.2.2 Femtosecond Stimulated Raman Spectroscopy

It is possible to modify the Impulsive Spontaneous Raman technique combin-ing a picosencond narrowband pulse and a broadband femtosecond pulse, in order to get a time-resolved snapshot of the non-steady state vibrational spec-trum, and to “overcome” the Heisenberg principle [14]; in fact, this basic layout disentangles the spectral and the temporal resolution of the experiment.

This technique is called Stimulated Raman Scattering (SRS), and will be discussed in detail later.

Adding a femtosecond photochemical pump turns the SRS in Femtosecond Stimulated Raman Scattering (FSRS, [23, 24, 25]).

This technique can be used to achieve the vibrational spectra of the excited state during the whole dynamic, but the Rayleigh scattering overlaps the low frequency vibrational peaks.

A schematic representation of a typical FSRS experiment is reported in Figure 2.5.

Some high-frequency vibrational modes in hemeproteins show an interest-ing time evolution, that is caused by the anharmonicities of the system [27] (Figure 2.6).

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2.2. ULTRAFAST VIBRATIONAL EXPERIMENTS 11

Figure 2.4: Spontaneous Raman experiments on hemeproteins. The vibrational peak near 200 cm−1 is a signature for the presence of the ligand. However, the impulsive spontaneous Raman shows its limits when investigating the sub-picosecond vibrational dynamics.

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Figure 2.5: Schematic representation of a typical FSRS experiment. Three laser pulses are focused on the sample. The first one, the actinic pulse, excites the system and triggers the dynamic of interest. Then the vibrational modes of the excited state are probed with two pulses, the Raman and the probe. The Raman is a picosecond pulse, with a narrow spectrum, that fixes the frequency resolution of the technique, while the probe is a broadband ultrashort pulse, that triggers the time when the sample is probed [26].

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(b) ν4

Figure 2.6: FSRS experiment on the hemeprotein. The spectrum measured with FSRS at different time delays with the actinic. This shows a strong improvement from the impulsive spontaneous Raman (Figure 2.4). The mode at 1355 cm−1 (the ν4)

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The Rayleigh scattering prevents the FSRS to study the dynamic of the 200 cm−1 _{mode, of great interest in hemeproteins as it is a signature for the}

ligand-presence.

To overcome this inconvenient a novel technique, the Impulsive vibrational spectroscopy (IVS), has been introduced.

2.2.3 Impulsive Vibrational Spectroscopy

IVS is the upgraded version of the TA experiment and involves two ultrashort beams1_.

The first one, the actinic pump, excites the sample, then the second beam probes the system at different times. When the actinic pump is sufficiently broadband and time compressed it can excite more vibrational modes of the same electronic level at once, producing a localized wavepacket that oscillates coherently. If the two pulses are sufficiently short in time it is possible to resolve the coherent oscillation of the wavepacket evolving in the excited state.

The signal of this kind of experiment is resolved in time, and needs to be Fourier transformed to get the vibrational spectrum (Figure 2.7).

Figure 2.7: A schematic representation of the IVS. The vibrational coherence is generated by the actinic pulse on the excited state, and probed by the probe pulse (a). The signal is in time domain, to get the spectrum it must be Fourier transformed (b). Adapted from [6]

In this experiment low frequency modes oscillate slower, and are easier to see than high frequency modes. This technique is somehow complementary to the FSRS.

Figure 2.8 shows the coherence oscillation of the low frequency part of the hemeproteins spectrum, obtained with a 70 fs actinic pulse.

An high compressed actinic pulse is the key to the measurement of good IVS spectra. Using ≈ 10 fs pulses make it possible to get even the high frequency mode using IVS (Figure 2.9).

Critically, both FSRS and IVS suffer of the ground state background, pro-duced by the remaining population on the ground state. In fact the actinic does 1_{The FSRS experiments involve at last three pulses, the actinic followed by the two pulses}

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Figure 2.8: The low frequency vibrational spectrum achievable using a 70 fs actinic pulse. Adapted from [28].

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not excite the whole sample, so the signal produced by the electronic excited state is superimposed with that one produced by the ground state.

Separating these two contributes is one of the most important challenges of modern ultrafast spectroscopy and it is one of the focus of this work.

In a recent work Kukura et. al. showed how to correct this unwanted spectrum by using a third beam that suppresses the non resonant background in IVS [30].

Also the FSRS ground-state background can be suppressed tuning the reso-nance of the SRS with an other excited state of the system, but both techniques are system specific, and cannot be applied to all possible systems.

Resonances between the probe pulse with different electronic excited states is a way to enhance the signal in both techniques; it is possible to exploit this mechanism to isolate the signal produced only by one excited state.

Since the pioneer works of Ruhman et. al. [31] the role of pulse shape in the detected signal has been investigated for the resonant case.

In this thesis new implications of the pump and probe shapes have been investigated, with a particular focus on the off-resonant processes where there is no electronic transition, nor excited dynamic to be taken into account for the signal shape, and more complex effects play together to create the signal.

These effects are of course present also in the resonant case, of a wider interest, but have been neglected in the standard theoretical modelling of IVS signals.

Also, in the off-resonant case, a complex dependence of the mode intensity form the probe wavelength has been observed and not correctly explained by the current theoretical description (Figure 2.10).

Figure 2.10: The intensity of two modes of the cyclohexane (C6H12) as function

of the probe wavelength. As can be seen the 800 cm−1 vibrational mode shows a bilobed profile, with a minimum on the central probe wavelength, while, at the same wavelength, the 384 cm−1 mode shows a maximum.

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2.2. ULTRAFAST VIBRATIONAL EXPERIMENTS 17 The focus of this work is to provide a new theoretical model of the IVS that can successfully describe how the experimental parameters (as pulse chirp [32], group velocity dispersion inside the sample, and phase-matching) affect the mea-surement, to spotlight which phenomena can significantly change the spectrum measured, even enhancing or suppressing some vibrational peaks.

This would permit to act directly on pulse shaping to perform a selective mode spectroscopy, focused on following the dynamic of single low intensity mode enhancing its signal.

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Chapter 3

Quantum theory of

nonlinear spectroscopy

In this chapter the quantum theory of the interaction between electromagnetic fields and matter is developed, and then used to predict the signal in ultrafast optics techniques. The second quantization theory has the advantage that can be applied even to study multidimensional spectroscopy with entangled light [33, 34].

3.1 The interaction Hamiltonian

The complete hamiltonian of the electromagnetic filed and matter can be sepa-rated into three parts:

H(t) = H0(matter)+ H (em)

0 + Hi(t)

In the Coulomb’s gauge this Hamiltonian has the following form [10, 35, 36]: H = 1 2m X α h ~_P_α_{+ e ~}_A(~r_α₎i2 +1 2 Z drσ(~r)φ(~r) +1 2 Z dr ε0E~T2(~r) + 1 µ0 ~ B2(~r) (3.1) Here the standard kinetic term is modified to take into account the presence of a magnetic field:

~

P → ~P+ e ~A

The form of the electrostatic coupling between atomic charges, using the scalar potential in the Coulomb’s gauge is:

φ(r) = 1 4πε0 " −X α e |~r − ~rα| +Ze r #

And the atomic density charge can be expressed as: σ(~r) = −eX

α

δ(~r − ~rα) + Zeδ(~r)

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This term represents the free matter Hamiltonian, that is not coupled with electromagnetic field: H₀(matter)= 1 2m X α P_α2+1 2 Z drσ(~r)φ(~r)

The last integral is the free electromagnetic Hamiltonian, in which electric and magnetic fields operators are not coupled with matter.

H₀(em)=1 2 Z dr ε0E~T2(~r) + 1 µ0 ~ B2(~r) (3.2) The remaining terms inside the first term are the electromagnetic-matter coupling HI mc= 1 2m X α h

−e ~Pα· ~A(~rα) − e ~A(~rα) · ~Pα+ e2A(~r~ α) · ~A(~rα)

i

(3.3) This is the interaction Hamiltonian in the Coulomb Gauge, and is commonly known as minimal coupling Hamiltonian. However it is quite difficult to develop a good perturbation theory: the involved field operator is the potential vector, which is not an observable quantity, and so it is not directly related to the experimental measurements, and it depends on the gauge. The vector poten-tial appears quadratically, which leads to a more complex form of the optical response.

To solve these issues we must perform a canonical transformation, introduced by Gr¨oppert-Mayer [37], Power and Zienau [38].

The transformation takes place defining a unitary operator: U = exp i

~ Z

dr ~P(~r) · ~A(~r)

Where ~P is the polarization operator, that can be expressed as: ~ P(~r) = −eX α ~rα Z 1 0 dξδ(~r − ξ~rα) (3.4)

This form is obtained directly by the classical definition of polarization vec-tor, and its direct calculation is reported in [35].

The new Hamiltonian and wavefunctions are obtained by the means of a unitary transformation:

H = U+_H

mcU |ψi = U+|ψoldi

Under this transformation the electric field becomes: U+E~TU = ~ET − 1 ε0 ~ P = 1 ε0 ~ D (3.5)

Substituting into equation (3.2) we obtain: 1 2 Z dr ε0ET2(~r) + 1 µ0 B2(~r) − Z dr ~P(~r) · ~ET(~r) + 1 2ε0 Z drP2(~r)

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3.1. THE INTERACTION HAMILTONIAN 21 The first part is again the free electromagnetic Hamiltonian. The last term involves only the matter. The new interaction term is given by the central relation.

The new Hamiltonian is no more expressed in terms of the vector potential, but in terms of electric and magnetic fields. Both the ~A · ~P and the A2 _terms

transform involving only magnetic contributions. Since in optical applications the electric field’s interactions are much stronger than the magnetic interaction, these last contributions can be neglected. So the transformed Hamiltonian of the interaction in minimal coupling Hamiltonian is zero.

The final transformed Hamiltonian appears as follows: H0= 1 2m X α P_α2+ Z drσ(~r)φ(~r) + Z drP 2 2ε0 +1 2 Z dr ε0ET2(~r) + B2_(~r) µ0 HI = − Z dr ~P(~r) · ~ET(~r)

This interaction term is easier than the interaction of the minimal coupling Hamiltonian, and it is a good starting point to develop the perturbation theory. Now we can proceed expanding the polarization in terms of the displacement vector ~D defined in equation (3.5). Using the definition of P in equation (3.4) we can rewrite this interaction term:

HI = e Z drX α Z 1 0 dξδ(~r − ξ~rα)~rα· ~ET(~r)

The first integral can be performed using the δ function: HI = e X α Z 1 0 dξ~rα· ~ET(ξ~rα)

Now the electric field can be expanded in Taylor series of ξ: HI = e X α Z 1 0 dξ 1 + ξ~rα· ~∇ + 1 2! ξ~rα· ~∇ 2 + · · · ~ rα· ~ET(0)

This is the multipole expansion. Since the order of magnitude of each of this terms is related to the power of the fine structure constant, equal to 1/137, we can easily consider only the first term, neglecting the others. This is commonly known as electric dipole approximation, and the Hamiltonian reads as follows:

HI = e

X

α

~rα· ~ET(0) = ~µ · ~ET(0) (3.6)

Where ~µis the dipole moment operator.

3.1.1 Second quantization of the atomic Hamiltonian and

Electric field

Since now we have considered the atomic Hamiltonian in the first quantiza-tion formalism, in which operators like particle posiquantiza-tion and momentum apply directly to the wave function of the system.

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Let us suppose we have solved the atomic Hamiltonian. Then it can be expressed as a function of its eigenstates |ni:

Hmatter=

X

n

~ωn|ni hn|

Where ~ωn is the eigenvalue of the |ni eigenstate. Since Hmatteris an hermitian

matrix, its eigenstates form an orthonormal basis for the Hilbert space. Using the definition of the Identity:

I=X

n

|ni hn| We can rewrite the Hamiltonian:

Hmatter=

X

i

|ii hi| Hmatter

X

j

|ji hj| =X

i

~ωi|ii hi|

The same transformation can be performed for all operators. Here we are interested in the dipole moment operator:

~

µ=X

i,j

|ii hi| ~µ |ji hj|=X

i,j

hi|eX

α

~rα|ji |ii hj|

This represents a transition operator between the atomic states. When ~µ is applied to a state, it transforms the state into a linear superposition of any other different state, with transition amplitude related to ~µij.

At the same time the electric field can be subdivided into two contributes: ~

E(~r, t) = ˆX

j

Ej(~r, t) + Ej+(~r, t)

Where ˆrepresent the polarization versor, E and E+_{the positive and negative}

oscillation frequency of the j-th mode Ej = r ~ωj 2ε0V aje−i(ωjt−~kj·~r) (3.7) E_j+= r ~ωj 2ε0V a+_jei(ωjt−~kj·~r) _(3.8)

Where aj and a+j operators are the annihilation and creation operators for

the j-th photon’s mode.

Since all the nonlinear optical techniques treated in this thesis do not use the polarization of the incoming fields, we include the scalar product between ˆ and ~µin the definition of µ so that

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3.1. THE INTERACTION HAMILTONIAN 23 and the vector symbols in the following equations1_{can be omitted for simplicity.}

Now we have defined all the quantities in the interaction Hamiltonian using the second quantization formalism, and it is expressed as:

HI = X ij i6=j µij|ii hj|Ej(t) + Ej+(t) (3.9) Here we have separated the photon emission and annihilation in two complex conjugate parts of the Hamiltonian, we can do the same thing for the excitation and decay of the molecule:

V = X

ij j>i

µij|ii hj|

So the interaction Hamiltonian is: HI = V + V+ X j Ej+ Ej+ (3.10)

3.1.2 Optical response function

The final target of this work is to evaluate the optical response function. Ex-perimentally the laser beam spectrum is monitored after it has interacted with the sample and it can be defined as:

S0= hNp(tf)i − hNp(0)i

Np denotes the number of photons on the mode of the probe beam. The signal

here is the difference between the number of photons evaluated after and before the beam has passed trough the sample. S0 _{can be recasted in the form:}

S0=

Z _{d hN}

pi

dt dt The term that we want to evaluate is:

S=d hN ip dt

This time derivative can be evaluated using the Ehrenfest’s theorem2_:

S= i

~[HI, Np]

Np= a+pap (3.11)

Eq. (3.11) can be explicity solved using the commutator rules between bosonic creation and annihilation operators:

[ai, aj] = [a+i , a +

j] = 0 [ai, a+j] = δij (3.12)

1_{There are some experimental techinques, like RIKE (Raman induced Kerr effect), that}

exploit the tensorial form of the non linear response function of materials; for those it is convenient to maintain the vectorial notation for both electric fields and the dipole moments all the perturbation theory, but this would be pedant for our purpose.

2_{The Hamiltonian in the commutators of equation (3.11) should be the complete}

Hamil-tonian. However, the Npoperator commutes with both the non interactive Hamiltonians, so

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[HI, a+pap] = X j V Ej+ V Ej++ V +_E j+ V+Ej+, a + pap

As the dipole operator commutates with a+

sascomutator we can take it outside:

[HI, a+pap] = (V + V+) X j [Ej, a+pap] + [Ej+, a + pap] = = [HI, Np] = (V + V+) X j [Ej, a+p]ap+ a+p[E + j , ap]

Combining equations (3.7, 3.8) with equation (3.12) we get: [HI, Np] = (V + V+) Ep− Ep+ S= i ~(V + V +_{) E} p− Ep+ (3.13) So the signal can be measured evaluating the expectation value of the oper-ator in eq. (3.13). Using the Schr¨odinger approach, which consists in evaluating the time evolution of the wavefunction, the expectation value of the observables is computed directly using the solution of the Schr¨odinger equation:

hAi = hψ(t)|A|ψ(t)i

However, this approach is not well suited for systems where T 6= 0, in which decoherence phenomena take place, and the real expectation value must be evaluated by averaging on a mixture of possible states:

hAi =X i pihψi|A|ψii X i pi= 1

For this reason it is useful to introduce the density matrix operator ρ [39] defined as

ρij = |ψii hψj|

It is straightforward to prove that the expectation value is: hAi =X

i

(ρA)_ii = Tr [ρA] and the signal can be expressed as:

S = Tr i ~(V + V + ) Ep− Ep+ ρ(t) (3.14) S =2 ~= Tr V E + pρ(t) + = Tr V +_E+ p ρ(t) (3.15)

3.2 Time evolution of the density matrix

Computing the signal in equation (3.15) requires to find an expression for the time evolution of the density matrix.

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3.2. TIME EVOLUTION OF THE DENSITY MATRIX 25 ˙ρij = d |ψii dt hψj| + |ψii d hψj| dt Using the Schr¨odinger equation for the wavefunction3_:

˙ρij= −

i

~(Hρij− ρijH) = − i

~[H, ρij]

This equation can be written in the Liouville space: the Liouville space is a standard Hilbert space in which operators are vectors, and the “operators” in the Liouville space (called superoperators) act on the standard operators of the Hilbert space.. Introducing the Liouville superoperators:

HLρ= Hρ HRρ= ρH

[H, ρ] = (HL− HR)ρ

Defining the superoperator H− = HR− HL √ 2 H+= HR+ HL √ 2 The density matrix evolution in the Liouville space is:

˙ρ = −i ~

√ 2H−ρ

Or in a more compact way:

˙ρ = −i ~Lρ

Here L has the same role of the Hamiltonian of the system in the standard Hilbert space. The evolution of the density matrix can be expressed as a unitary transformation performed by a superoperator U defined as:

ρ(t) = U(t, t0)ρ(t0) U (t, t0) = exp −i ~ Z t t0 L(t0)dt0

However, this expression is not easily evaluable, since L is composed by both interactive and non interactive terms that do not commute each other, so this exponential cannot be directly applied to the density matrix.

3.2.1 Interaction picture in the Liouville space

As the Liouville operator L is a linear combination of the hamiltonian super-operators it can be expressed as the sum of a non interactive operator and an interactive operator:

L = L0+ LI

LI =

√ 2HI −

We can define the interaction scheme making a unitary transformation that depends on time, using the superoperator U0= e−

i ~L0t.

3_{The time evolution of the density matrix has a minus sign with respect to the Heisenberg}

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Superoperators in the interaction picture evolve in the following way AI = U0ASU0+

While the state of the system (the density matrix) is obtained: ρI(t) = U0+ρS(t) = e i ~L0tρ_S(t) ρI(t) = e i ~L0te− i ~Ltρ_S(0)

Now let us get the time evolution equation for the density matrix in the interaction picture: ˙ρI(t) = i ~ L0SρI(t) − i ~ ei~L0t(L_0S+ L_{I S}) e− i ~LStρ_S(0)

In the last equation all the superoperators are expressed in the Schr¨odinger picture. ˙ρI(t) = i ~L0SρI(t) − i ~L0SρI(t) − i ~e i ~L0tL_{I S}e− i ~L0te i ~L0tρ_S(t) ˙ρI(t) = − i ~L 0 IρI(t) (3.16) Now L0

I is computed in the interaction scheme.

We can solve this equation integrating both sides of equation (3.16): ρI(t) = ρI(t0) − i ~ Z t t0 dt1L0I(t1)ρI(t)

This is a iterative equation, that can be solved: ρI(t) = ∞ X n=0 −i ~ nZ t t0 dt1 Z t1 t0 dt2· · · Z tn−1 t0 dtnL0I(t1) · · · L0I(tn)ρI(t0) = X n ρ(n)_I (t) The problem of the convergence of this series has been examined well in the last century [40]. We focus only to a finite order perturbation theory, studing phenomena only at fixed order.

Transforming to the Shr¨odinger picture we obtain: ρ(n)(t) = U0(t)ρ (n) I (t) ρ(n)(t) = −i ~ nZ t t0 dt1· · · Z tn−1 t0 dtnU0(t)U0+(t1)LIU0(t1) · · · U0+(tn)LIU0(tn)U0+(t0)ρ0

Where we have used:

ρI(t0) = U0(t0)ρ0

Recalling that the U+

0 can be seen as a backward evolution in time:

U0(t2)U0+(t1) = U0(t2, t1) We have: ρ(n)(t) = −i ~ nZ t t0 dt1· · · Z tn−1 t0 dtnU0(t, t1)LIU0(t1, t2)LI· · · LIU0(tn, t0)ρ0 (3.17)

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3.2. TIME EVOLUTION OF THE DENSITY MATRIX 27 This expression is very useful for calculation, and has also a readable form: the density matrix interacts n times with the Liouville interaction operator, and between these interactions it evolves following the non interactive time evolution operator.

In particle or condensed matter physics the time ordering operator would be now introduced and the result wuold be expanded using the Wick’s theo-rem. However in ultrafast optics the interactions are time resolved4 _{and it is}

convenient to keep the time ordering of the integrals.

Now we can insert this expression inside the signal equation (3.15) to get the full signal. Let us call the first term in the equation (3.15) Srwa:

S_rwa(n) = 2 ~ = Tr V E_p+ −i ~ nZ t t0 dt1· · · dtnU0(t, t1) · · · U0(tn, t0)ρ0 S_rwa(n) = 2 ~ = −i ~ nZ t t0 dt1· · · dtnTrV Ep+(t)U0(t, t1)LI(t1) · · · ρ0 (3.18) And the second term:

S_nrwa(n) = 2 ~ = −i ~ nZ t t0 dt1· · · dtnTrV+Ep+(t)U0(t, t1)LI(t1) · · · ρ0 (3.19)

3.2.2 Non interactive evolution in the Liouvile space

The signal equation (3.18) and (3.19) contains the free evolution of the density operator in the Liouville space. The density matrix is expressed as the tensorial product of the atomic eigenstates and a field basis.

The laser beam can be considered as coherent light, whose states are eigen-states of the electric field:

E(r, t) |ψi = E(r, t) |ψi

Where E(r, t) is the classical electric field associated to the |ψi laser light. For this reason the semi-classical approach is well suited to describe optical nonlinear process. When the light states of the density assume only coherent light states, then the semi-classical description is equivalent to the second quan-tization result. However if the system interacts with quantum light, or with the vacuum field (n = 0 photons), they are not eigenstates of the electric field operator. So the complete second quantization technique is required.

The time evolution of coherent states is not so trivial to study since they are not eigenstates of the quantum electromagnetic Hamiltonian. However, the dynamic of such states become relevant only when the laser light contains only a small number of photons. In the current applications we are using a very intense laser source, and the quantum uncertainty related to such states can be neglected. This can be easily demonstrated as follows.

The coherent state is represented as: |αi = e−12|α| 2 ∞ X n=0 αn √ n!|ni

4_{The interaction between fields and matter can happen only when the laser pulse arrives}

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This definition is so that it is an eigenstate of the destruction operator. a |αi= α |αi

Introducing the displacement operator [41], that, acting on the vacuum state, displaces it in the coherent state:

D(α) |0i = |αi D(α) = eαa+−α∗_a

It is easy to check that this is a unitary operator: D+(α)D(α) = D(α)D+_{(α) = I}

And it satisfies the following rules:

D+(α)aD(α) = a + α D+(α)a+_{D(α) = a}+_{+ α}∗

Where a+ _{and a are the creation and annihilation photon operators. It is}

straightforward to prove, if the laser light is very intense (|α|2_{= hni 1), that}

the coherent light becomes stationary for the Hamiltonian. H₀(em)=X

j

~ωja+jaj

a+a |αi= a+_{α |αi}_{= αD(α)D}+_(α)a+_{D(α) |0i =}

= αD(α)a+_{+ α}∗_{|0i =}

= αD(α) |1i + |α|2_|αi

In the limit α → ∞, the first term can be neglected5_{, and it can be considered}

an eigenstate of the free electromagnetic Hamiltonian.

This approximation is equivalent to the semiclassical approximation, since the state αD(α) |1i is not a coherent state, and takes into account for the sponta-neous emission when interacting with the sample in the interaction Hamiltonian. More interesting is what happens to the matter density matrix during the unperturbed evolution.

Considering a molecule composed by a two level system, a ground state |gi and an electronic excited state |ei. The density matrix is a 2x2 matrix.

ρ=ρgg ρge ρeg ρee

(3.20) The diagonal elements of the density matrix are called population elements, and represent the probability of finding a molecule in the ground or in the excited state. The off-diagonal elements are called coherent elements, and describe the probability of finding the system in a linear superposition of eigenstates.

A pure classical system is when the density matrix has only diagonal terms. This state does not correspond to any wavefunction, but is a mixture of wave-functions, that does not produce interference phenomena.

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3.2. TIME EVOLUTION OF THE DENSITY MATRIX 29 With this basis all the elements of ρ evolve stationary.

˙ρij = − i ~ L0ρij ˙ρij = − i ~ H_L(0)ρij− H (0) R ρij ˙ρij = − i ~(Ei − Ej) ρij(t) = e− i ~(Ei−Ej)tρ_ij(0) = e−iωijtρ_ij(0)

This evolution takes place only in system at T = 0. In real system decoher-ence phenomena occur, such that coherdecoher-ence terms i 6= j decay in time.

Decoherence can be inserted empirically in the Liouville operator: L0= HL− HR− iΓ

So that the real evolution of the system becomes:

ρij(t) = e−iωijt−Γijtρij(0) (3.21)

It is interesting to notice that the insertion of the dephasing term Γ inside the Liouville operator cannot be done easily in the wavefunction formalism. So this is one of the advantage of working in the Liouville space.

3.2.3 Empirical description of the dephasing.

We must add the dephasing in the Luoville operator, I will show the result of a simple simulation to better understand what happens during a dephasing process. This decription have been firstly proposed by Kubo [42].

Let us suppose that all the molecules of the system start oscillating due to a laser excitation at the same phase. For T 6= 0 in the presence of a bath, the surrounding will constantly push and pull at the molecule, and cause a stochastic force on the molecule:

ω(t) = ω0+ δω(t) (3.22)

And

hω(t)i = ω hδω(t)i = 0

Where the averages are intended both as time and ensamble averages. The dypole moment of the single molecule in absence of bath is a standard harmonic oscillator:

d

dtµ(t) = −iωµ(t) That yields the solution:

µ(t) = µ0e−iωt

Considering the bath (eq. 3.22) the solution is the result of the integral: µ(t) = µ0e

−iRt t0ω(t

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Meaning over all the molecules in the system we get: hµ(t)i = µ0e−iω0the−i

Rt 0δω(t

0_)dt0

i

A analytical computation of the ensample average can be performed using the cumulant expansion.

Figure 3.1 shows the result of a simulation with N = 1000 harmonic oscil-lators starting in phase, and then colliding randomly with a rate of 10 per unit of time, and changing their phase by 0.1 radians each time.

Figure 3.1: Simulation of N = 1000 harmonic oscillations, that vary their phase ran-domly during the evolution. The mean of all the positions of the harmonic oscillators is shown in the top plot, together with two particular oscillators.

The resulting oscillation is dumped in time, even if all the single oscillations maintain the same amplitude.

3.3 Static absorption

As an example we will integrate the first order optical signal. Let us consider the two level system for atomic Hamiltonian, described by the density matrix in equation (3.20).

The measured optical signal is given by equation (3.18). S_rwa(1) = 2 ~ = −i ~ Z t t0 dt1TrV Ep+(t)U0(t, t1)LI(t1)U0(t1, t0)ρ0 S(1)_nrwa= 2 ~ = −i ~ Z t t0 dt1TrV+Ep+(t)U0(t, t1)LI(t1)U0(t1, t0)ρ0

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3.3. STATIC ABSORPTION 31 We can rearrange the signal as:

S(t) = 2 ~ =V E+ p(t)P (t) P(t) = −i ~ Z t −∞ dt1U0(t, t1)LI(t1)U0(t1, t)

This is the signal function if the detection is completely time resolved, where the probe spectrum operator is convolved with the polarization with the detector time resolution function:

S(t) = Z dt02 ~ =hV E+ p(t)P (t0)i δ(t − t0)

In this case the detector can perform a time resolved measurement on the probe (that is represented by the Dirac delta funciton).

A more interesting case is when the signal is frequency resolved. In this case it is more convenient to write the response function directly in the frequency domain: S(ω) = Z dω02 ~=hV E + p(ω)P (ω0)i δ(ω − ω0) S(ω) = 2 ~=V E + p(ω)P (ω) (3.23) This is of course completely different from the Fourier transform of the time signal, since the detector responce function is a Dirac delta function in fre-quency [10]. In the latter calculation we will always use equation (3.23), since all the experiment that we will take into account are frequency resolved.

Now the density matrix evolves through t0→ t1without interacting with the

electric field. Since it starts from a population state in the ground level, there is no evolution at all. Then it interacts with LI(t1). We have two interaction

terms: LI = H (int) L − H (int) R

The interaction can happen both on the left side (with a + sign) or on the right side of the density matrix. This will give two contributions.

HLρ − HRρ

Where:

H(int)= (V + V+_{)(E + E}+_{) = V E + V E}+_{+ V}+_{E + V}+_E+

So we have eight terms (4x2) in this first interaction, and two signals to evaluate, for a total of 16 different terms. However, it is easy to see that most of them are zero. For example, before the first interaction the system is in the ground state, so when V+ _{acts on the density matrix it destroys the signal.}

V E ρ −ρV E V E+ρ −ρV E+

We can apply these operators on the density matrix ρ0= |gi hg| ⊗ |Ep(t)i hEp(t)|

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µgeEp(t1) |ei hg| − |gi he| Ep(t1)µge

µgeEp∗(t1) |ei hg| − |gi he| µgeE∗p(t1)

After this step we have a time evolution, from t1 to t. The density matrix is

not in a population state, so the evolution follows equation (3.21).

µgeEp(t1)e−iωeg(t−t1)−Γge(t−t1)|ei hg| − |gi he| eiωeg(t−t1)−Γge(t−t1)Ep(t1)µge

µgeEp∗(t1)e−iωegt−Γget|ei hg| − |gi he| eiωeg(t−t1)−Γge(t−t1)µgeEp∗(t1)

Then the operators are applied on the left side of the density matrix. How-ever, again since this is a two level system, only one dipole operator does not destroy the signal. Since we are using the equation 3.23, this time the electric fields enters in the frequency domain.

µ2

geEp∗(ω)Ep(t1)e−iωeg(t−t1)−Γge(t−t1)|gi hg| − |ei he| eiωeg(t−t1)−Γge(t−t1)Ep∗(ω)Ep(t1)µ2ge

µ2

geEp∗(ω)Ep(t1)∗e−iωegt−Γget|gi hg| − |ei he| eiωeg(t−t1)−Γge(t−t1)µge2 Ep∗(ω)Ep∗(t1)

Now we have to evaluate the trace. Remarkably all the modes are in a population state, so the trace corresponds to the sum of all the coefficients of the density matrix.

Critically, with more than two states, pathways where the density matrix ends on a coherent state exist. These terms give zero contribution to the signal, since the signal is generated by the trace.

Now we must perform the integral over t1, and Fourier transform:

P1(ω) = −i ~ µ2_ge Z ∞ −∞ dteiωt Z t −∞ dt1Ep(t1)e−iωeg(t−t1)−Γge(t−t1) P2(ω) = − −i ~ Z dteiωt Z t −∞ dt1eiωeg(t−t1)−Γge(t−t1)Ep(t1)µ2ge P3(ω) = −i ~ µ2ge Z dteiωt Z t −∞ dt1Ep∗(t1)e−iωegt−Γget P4(ω) = − −i ~ Z dteiωt Z t −∞ dt1eiωeg(t−t1)−Γge(t−t1)µ2geE ∗ p(t1)

Let us perform explicitly only first integral: P1(t) = −i ~ µ2_ge Z t −∞ dt1Ep(t1)e−iωeg(t−t1)−Γge(t−t1)

Performing a change in variable τ = t − t1

P1(t) = −i ~ µ2ge Z ∞ 0 dτ Ep(t − τ )e−iωegτ −Γgeτ Defining Ep(t) = Z ∞ −∞ ˜ Ep(ω1)e−iω1tdω1

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3.4. FEYNMAN RULES IN TIME DOMAIN 33 P1(t) = −i ~ µ2ge Z ∞ −∞ dω1E˜p(ω1) Z ∞ 0 dτ e−iω1(t−τ )_e−iωegτ −Γgeτ

The integral in τ can be performed easily: Z ∞

0

dτ e−iω1τ_e−iωegτ −Γgeτ ₌ 1

−i(ω1− ωeg+ iΓeg)

If we want to take the spectrum of the signal we must perform the Fourier transform: P1(ω) = µ2 ge ~ Z ∞ −∞ dteiωt Z ∞ −∞ dω1E˜p(ω1)e−iω1t ω1− ωeg+ iΓeg

The integral in t can be performed easily and takes out a delta function: δ(ω1− ω) ω1= ω

This corresponds to the conservation of the energy during the interaction, and it is a common feature of the perturbation expansion in the Fourier’s space [43]. P1(ω) = µ2_ge ~ · Ep(ω) ω − ωeg+ iΓeg

Then substituting inside equation (3.23) we get the observed signal: S(ω) = 2µ 2 ge ~2 = _E∗ p(ω)Ep(ω) ω − ωeg+ iΓeg S(ω) = −2µ 2 ge ~2 Ip(ω) Γeg (ω − ωeg)2+ Γ2eg (3.24) The signal is negative. This means that the photons at the end of the sample are less than the incoming photons. This is obvious since we are studying the static absorption. The sample absorbs more photons in the region of the probe where ω = ωeg, that is the resonance condition. And the absorption has a

line-shape given by the dephasing between the electronic excited and the ground state.

The signal is proportional to µ2

ge that is a measure of the spatial overlap of

the wavefunctions of the ground and the electronic state, since: µge∝ hg|~r · ˆ|ei

In principle we have the contribution from all the other quantum pathways; however, we can neglect them. It is possible to represent in a graphical way each quantum pathway (Feynman’s diagrams) and to directly evaluate the integrals corresponding at each pathway from the diagram. In the next section the general rules for constructing Feynman’s diagrams on these processes are explained, and then the integral is performed directly for all the remaining terms.

3.4 Feynman rules in time domain

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• We represent the time evolution of the density matrix with two lines (the bra side and the ket side).

• Each time that the interaction Liouville’s operator acts on the density matrix, an arrow is drawn on the left or right side of the diagram. • If the interaction takes place with the operator E+ _{it is a right arrow, a}

left arrow otherwise.

• Between the drawn arrows we must specify the current status of the den-sity matrix.

• The last arrow is always a left arrow that acts on the left side of the density matrix (the last interaction is V E+

p on the left side of the density matrix).

• The last interaction have to leave the density matrix in a population state (otherwise the trace of the matrix will be zero)

By these rules each different integral can be simply recasted in a graphical representation.

Given these rules it is possible to draw all the diagrams corresponding to the first order process computed in section 3.3; they are reported in Figure 3.2. Obviously it is possible to rewrite the integral from the digram, following the inverse rules:

• An arrow we represents the classical electric field (or its complex conjugate if the arrow is left directed).

• Each diagram gives an overall sign (−1)n_{, where n is the number of}

inter-actions on the bra side: that is because the Liovuille operator is LI = HL− HR

• During each interval between two arrows the signal have to be multiplied by the time evolution phase factor of the current density matrix:

e−i˜ωijτ _ω_˜

ij= ωi− ωj− iΓij

Where τ is the time interval between the two arrows.

• For each arrow (even the last) the result must be multiplied by the dipole moment of the corresponding density matrix transition.

• The result must be multiplied by a factor − i ~

n

, where n is the pertur-bative order.

So the integral of the second diagram it is: S(t) = −2µ 2 ge ~ = −i ~ E_p∗(t) Z ∞ 0 dτ Ep(t − τ )e−i˜ωgeτ And so on.

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3.4. FEYNMAN RULES IN TIME DOMAIN 35

(a) 1 (b) 2

(c) 3 (d) 4

Figure 3.2: Diagrams representing the static absorption. The diagrams give us a better physical understanding behind the mathematical formulation of the theory. The first pathway is the only one who has a clear physical meaning. The first interaction is an absorption, and the state of the system is excited after this interaction. The second and the third diagrams have a not obvious meaning: the first interaction is an emission, but the system is excited. This interaction does not conserve the energy. However the conservation of energy is guaranteed in the second diagram by the last interaction.The third diagram does not conserve the energy globally, as the fourth. The last two diagrams cannot occur, while the second can occur, but its cross-section is very small compared to the first.

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3.4.1 Feynman rules in Fourier Space

Feynman diagrams can be recasted also in the Fourier space. As we have shown in section 3.3, this is the common way to resolve experiments.

In this case we Fourier transform all the fields envelopes: Ep(t) =

Z ∞

−∞

dω1E˜p(ω1)e−iω1t

So that the polarization becomes: P(ω) = Z ∞ −∞ dteiωt Z ∞ −∞ ˜ Ep(ω1) Z ∞ 0 dτ e−iω1(t−τ )_e−i˜ωgeτ

The integral in τ is the evolution in the frequency space, and leads to the Green function:

G(ω1, ωge) =

1 −i(ω1− ˜ωge)

The integration on t brings us the Dirac’s δ function. δ(ω − ω1)

Which represents the energy conservation, and the final integral becomes (using a frequency resolved technique):

S(ω) = −2µ 2 ge ~ = 1 ~ E∗ p(ω)Ep(ω) ω −ω˜ge (3.25) So another diagram can be drawn to directly write this result.

Figure 3.3 shows the graphical representation of this diagram.

Figure 3.3: Frequency representation of the Feynman diagram of the second process in the static absorption.

We can now extrapolate the rules for evaluating integrals directly from the diagram:

• Each fields arrow have to be substituted with the field frequency envelope evaluated in a mute variable ωi. This frequency is added to the ”energy”

of the system if interaction is a right arrow, subtracted if it is a left arrow (and the field must be complex conjugate in this case).

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3.5. ROTATING WAVE APPROXIMATION 37 • For each interval between arrows a Green function of the energy of the system have to be evaluated. The energy of the system is obtained by summing the energy of all the previous interactions:

G(ωf ields, |ai hb|) =

1

ωf ields− (ωa− ωb− iΓab)

• Each diagram gives an overall sign (−1)n_{, where n is the number of}

inter-actions on the bra side.

• The corresponding dipole moment must be multiplied for each interaction between fields and the density matrix.

• The last interaction has frequency −ω (the detected frequency), and must conserve the energy (so one or more mute variables can be fixed). • The result is obtained integrating over all the remained mute variables. With these techniques all the results at the first order can be written directly from their corresponding diagram.

3.5 Rotating wave approximation

Since from the definition of the signal we have distinguished two contributions, the RWA and the NRWA.

That is because they form very different kind of diagrams: The NRWA signal represents a diagram in which the last interaction does not conserve the energy, it is a photon emission (E+_{) combined with a molecule excitation V}+_{. All the}

NRWA diagrams do not conserve the energy, at least in one of the interactions6

It is trivial to show that this diagrams have a small cross-section compared to the ”resonant” ones (where we define a resonant diagram a diagram in which all the interactions conserve the energy).

A clear example of the RWA diagrams against NO RWA ones is the first diagram vs the second diagram in static absorption. The two final signals are calculated in equations (3.24) and (3.25).

S1(ω) = − 2µ2 eg ~2 Ip(ω)Γeg (ω − ωeg)2+ Γ2eg S2(ω) = 2µ2 ge ~2 Ip(ω)Γge (ω + ωge)2+ Γ2ge

The first diagram near the resonance condition ω = ωeg is much higher than

the second one. That is because the field and the matter oscillate at opposed frequency (a decay combined with a photon emission or an excitation combined with a photon destruction).

So when the resonance condition occurs only the first diagram can be con-sidered. This is known as Rotating Wave Approximation (RWA).

However RWA can be used only when resonant transition can occur. Let us consider a two level system with an electronic transition far from the probe spectrum (the sample is transparent to the used probe). The signal must be 6_{The single interaction can violate the energy conservation law, but the overall energy must}

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equal to zero because the probe remains unchanged when passing through the sample.

In fact this is verified considering the sum between S1and S2:

Stot(ω) = 2µ2 geΓge ~2 Ip(ω) − 1 (ω − ωeg)2+ Γ2ge + 1 (ω + ωeg)2+ Γ2eg ωegω −→ 0 Since the main objective of this thesis is to correctly describe the off-resonant IVS signal, RWA cannot be considered in approximation.

3.6 Phase matching

Up to now we have neglected the spatial dependence of the fields inside the sample.

Considering a sample length L, the signal is generated while the beams propagate through, accordingly to:

S(ω) = 2 ~= " Z L 0 dzEp∗(ω, z)P (n) (ω, z) # Where: P(n)(ω, z) = −i ~ nZ ∞ −∞ dteiωt Z t −∞ Z L 0 dz1dt1· · · hU0(t, z, t1, z1)LI(t1, z1) · · ·i

Remarkably each interaction can take place in a different position inside the sample, and we must integrate over all possible positions.

However, if the system does not show high spatial quantum correlations (like superconductors) the interactions can be assumed to be local [10].

With this approximation all the interactions take place in the same place and

U0(t, z, t1, z1) = U(t, t1)δ(z, z1)

So only the last integral in z remains.

Let us apply this theory to the simulated static absorption (considering explicity the first diagram).

S(ω) = 2µ 2 ge ~2 = " Z L 0 dzIp(ω)e i[~ks(ω)−~kp(ω)]·~z ω −ω˜eg # S(ω) = 2µ 2 ge ~2 = _I p(ω) ω − ωeg+ iΓeg ei∆~k·~2L sinc ∆~k · ~L 2 !

The phase matching takes a phase factor that depends on ∆~k inside the signal, and multiplies it by an amplitude factor given by the sinc function.

The condition

∆~k = 0

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3.7. THIRD ORDER PROCESS: TRANSIENT ABSORPTION 39 In static absorption this condition is always true, because the measured field has the the same k-vector of the incoming field and so:

~_k_s_{= ~k}_p

However this is not always true for higher order processes.

The phase matching condition is another mechanism that can reduce the possible diagrams; in fact, when measuring the emitted signal along a specific direction, only the diagrams with ~∆k ≈ 0 will contribute to the total signal.

3.7 Third order process: Transient Absorption

The second order processes have a central role in generating ultrashort laser pulses at tunable wavelength. However they are possible only inside non cen-trosymmetric materials. So now we focus on third order processes, the lowest order used to exploit the ultrafast dynamics of common samples.

The simplest technique is the Transient Absorption (TA). As described in the introduction, the TA involves two short (hundreds of femtoseconds) laser pulses, the pump and the probe, that interacts with the sample at delayed time t0 [44].

Let us consider a three level system, one ground state |gi and two electronic states |ei₁ and |ei₂. We are interested in investigating a resonant third order process. We can apply RWA and neglect off-resonant interactions, and consider only phase-matched diagrams. We also assume that the transition between |gi and |e2i level is off-resonance with the excitation pump wavelength.

The transient absorption is the first time resolved technique, that has the advantage to be studied in the time domain (even if the detection is in the frequency domain).

Under these conditions, we have only six diagrams remaining, that we subdi-vide in three groups, Excited State Absorption (ESA, Figure 3.4), Ground State Bleaching (GSB, Figure 3.5) and Stimulated Emission (SE, Figure 3.6).

The signal can be computed in time domain, assuming the two electric fields to be instantaneous pulses.

Epu(t) =pIpuδ(t + t0) Epr(t) =pIprδ(t)

The two ESA diagrams describe a polarization of: P_ESA1(3) (ω) = − −i ~ 3Z ∞ −∞ dteiωt Z ∞ 0 dτ1dτ2dτ3Epu(t − τ1− τ2− τ3)Epu∗ (t − τ1− τ2)·

· Epr(t − τ1) exp [−i (˜ωe1gτ3− iΓe1τ2+ ˜ωe2e1τ1)] µ

2 e1gµ

2 e1e2

Where the minus sign takes into account the interaction on the bra side of the density matrix. Using the definition of Epu and Epr we have

τ3= 0 τ2= t + t0− τ1 τ1= t

Because of τ2>0 and τ1>0 we have two constraints on the final signal:

P_ESA1(3) (ω) = −IpupIprΘ(t0)e−Γe1t0 −i ~ 3Z ∞ −∞

dtei(ω− ˜ωe2e1)t_Θ(t)µ2

e1gµ

2 e1e2

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Figure 3.4: The diagrams that represent the ESA process. This is called ESA because the last interaction is an absorption while the system is populating the excited state.

Figure 3.5: The diagrams that represent the GSB process. The density matrix does not evolve trough an excited state between the two diagrams.

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3.7. THIRD ORDER PROCESS: TRANSIENT ABSORPTION 41

Figure 3.6: The diagrams that represent the SE process. The system completely absorbs the pump, then the probe performs stimulated emission.

Where Θ is the common Heaviside function. P_ESA1(3) (ω) = µ 2 e1gµ 2 e1e2IpupIpr ~ Θ(t0)e−Γe1t0 ω − ωe2e1+ iΓe2e1

The other diagram with the same fields produces the same result, because the only difference is in the first interval τ3, that has been fixed to 0 by the

impulsive interaction.

The complete signal for this diagram is Epr(ω) = 2πpIpr SESA(ω) = − 4πµ2 e1gµ 2 e1e2IpuIpr ~2 · Θ(t0)Γe2e1e −Γ_e1t0 (ω − ωe2e1) 2_{+ Γ}2 e2e1 (3.26) So the excited state absorption gives us a contribution that decays in time centered at the frequency of the optical transition between e1 and e2, and it is

a negative signal.

Using the same approach is possible to compute the GSB: SGSB(ω) = 4πµ4 e1gIpuIpr ~2 · Θ(t0)Γge1 (ω − ωe1g) 2_{+ Γ}2 e1g (3.27) GSB is a positive signal (gain) that does not depend on the dephasing of the excited state, centered at the resonance of the first excited state |e1i with the

ground state. SSE(ω) = 4πµ4 e1gIpuIpr ~2 · Θ(t0)Γe1ge −Γ_e1t0 (ω − ωe1g) 2_{+ Γ}2 e1g (3.28) The SE process has the same dependence on t0 of the ESA, but is centered

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is given by the sum of all these contributions sum of all this contributions, and the result is shown in Figure 3.7.

T A(ω) = SESA(ω) + SGSB(ω) + SSE(ω)

The parameters used in this simulation are:

Γe1g= 5 · 10 −4 _ps−1 _Γ e1 = 5 ps −1 _Γ e1e2= 4 · 10 −4 _ps−1 λe1g= 600 nm λe1e2 = 800 nm

Figure 3.7: Sum of all the signals of ESA, Bleach and SE. The loss (blue side) is the ESA, which is dominated by the BLEACH when t0→ ∞.

In a more complex framework it is possible to consider also a time dependent Hamiltonian for the molecule, so that when the electronic transition is activated, the system can perform non optical transitions, showing more complex signals that contain real information of ultrafast dynamics of the excited states [45]. This treatment is very system specific, and will not be deepened in this thesis.

3.8 Raman scattering

The TA alone does not provide information on the vibrational structure of the sample, as it involves only electronic transition. Raman overcomes this limit. In this section the Stimulated Raman Scattering is theoretically described. This

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3.8. RAMAN SCATTERING 43 technique can use off-resonant pulses, however we will use the RWA approxima-tion. This will be justified in the next chapter.

Even if historically spontaneous Raman has been developed far before the stimulated one, the complete theoretical description of spontaneous Raman in-volves spontaneous emission, and requires to abandon the classical limit of elec-tromagnetic field using second quantization formalism.

The SRS technique consists in two laser pulses, the Raman pulse, which is narrowband and has a time duration of few picoseconds, and the Stokes pulse, broadband and of few femtosecond.

Since the sample interacts twice with the Raman pulse and once with the Stokes, it is convenient to integrate directly the diagrams in the frequency do-main, and suppose the Stokes to be a monocromatic wave.

The two electric field are overlapped in time; no time ordering is present in this technique and in principle all possible diagrams must be taken under consideration. However, it can be shown that, in the off-resonant case, only two diagrams describe SRS signal [10, 46, 12], and thet are represented in Figure 3.8.

Figure 3.8: The two main diagrams in frequency domain of the Stimulated Raman Scattering. According to literature we will call the left one IRS (Inverse Raman Scat-tering), while on the right one RRS. Near each field arrow the interacting field is specified.

The signal can be easily written: SSRS(ω) = 2µ2 abµ 2 bc ~2 = Z _dω 1dω2Er∗(ω1)Er(ω2)Es(ω1− ω2+ ω)Es∗(ω) (−ω1− ˜ωab)(−ω2+ ω − ˜ωac)(ω − ˜ωbc) SIRS(ω) = 2µ2 abµ 2 bc ~2 = Z _dω 1dω2Er∗(ω1)Er(ω2)Es(ω1− ω2+ ω)Es∗(ω) (ω1− ω2+ ω − ˜ωba)(−ω2+ ω − ˜ωca)(ω − ˜ωba)

The two integrals can be performed using monocromatic Raman fields: Er(ω) = p Irδ(ω − ωr) SSRS(ω) = IrIs(ω) 2µ2 abµ 2 bc ~2 = 1 (−ωr− ˜ωab)(−ωr+ ω − ˜ωac)(ω − ˜ωbc)

In the off-resonant limit:

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Figure 3.9: Signal measured as Raman shift, the frequency shift of the peak position respect to the raman frequency.

SSRS(ω) Is(ω) ∝ = 1 ωr− ω − ωca− iΓca = Γca [ω − (ωr− ωca)]2+ Γ2ca (3.29) While the other leads to:

SSRS(ω) Is(ω) ∝ = 1 ω − ωr− ωca+ iΓca = − Γca [ω − (ωr+ ωca)]2+ Γ2ca (3.30) So in the red side of the spectrum we observe a positive peak, shifted by the vibrational frequency, while in the blue side we observe a symmetric loss (Figure 3.9).

If the Raman pulse was not monocromatic the result would be the spectral convolution between the Raman spectrum and the vibrational lineshape, loosing resolution as the spectrum became wider. The spectrum profile is not affected by the Stokes envelope. However to let the diagram exists the Stoke pulse must interact between the two Raman, so the diagram can occur only when both the beams interact with the sample, and the time resolution is reached by the shorter pulse.

This makes the SRS a powerfull probing technique for time resolved ultrafast experiment.

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Chapter 4

Modelling Impulsive

Vibrational Spectroscopy

As pointed out in the previous chapters, ultrafast spectroscopy relies on the study of the fast dynamic induced in the sample by a photoexcitation pulse. The transient absorption (TA) technique, described in Sec. 3.7, can resolve the electronic dynamic, but does not provide information on the structural changes of the molecules. The stimulated Raman scattering, described in Sec. 3.8, can perform measurement of the vibrational structure with high temporal resolu-tion; if combined with an actinic pulse it is a powerful technique to follow the vibrational dynamic of the sample.

However, the SRS suffers from the Reileigh scattering, that covers the low frequency region of the spectrum, making it useless to collect information con-tained in the low frequency modes.

To solve this issue the impulsive vibrational spectroscopy (IVS) has been developed. It consists in two femtosecond laser pulses: the use of broadband ultrashort pulses has the goal to excite vibrational coherence in the sample, and the signal in time shows oscillations modulated at the vibrational frequency (Fig-ure 4.1).

The IVS technique is usually performed when the laser beams are in reso-nance with an electronic transition, so the oscillations are superimposed with standard TA signal. However, both oscillations of vibrational levels in the ex-cited state and in ground state are observed. The development of an efficient way to discriminate them has been object of an intense debate ([47], [48], [49], [15]). In particular, [15] demonstrates that an appropriate chirp on the actinic pulse can be used to disentangle ground and excited state contributions.

The current modelling of IVS is performed using a semi-classical picture: the actinic generates a localized wavepacket on the excited state, that is evolved us-ing the Shr¨odus-inger equation. However this description holds only if the molecule has an excited state in resonance with the actinic pulse.

In this chapter we will focus on a more hidden technique, the off-resonant IVS, capable to measure the Raman active ground state vibrational levels of solvents where no excited state is in resonance with the laser pulses.

Since the off-resonant process does not involve any real electronic transition, the starting point of the broadband off-resonant IVS model is to turn down the

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Figure 4.1: The IVS experiment. On the left side (a) the experimental realization is presented: two pulses interact with the sample at delayed time, the signal is in the time domain, and the vibrational spectrum is obtained performing its Fourier transformation. On the right side (b) there is an intuitive representation of IVS; the vibrational coherence of the wavepacket is probed at different times.

RWA approximation.

4.1 Non RWA diagrams

Since we are considering all non-resonant transitions the RWA approximation does not hold any more. In principle additional diagrams are possible. In fact, in the off-resonant limit, we have eight surviving diagrams (instread of two), that are not cancelled by the time-ordering (the actinic interactions come before the probe ones), the energy conservation and the signal generation (in fact during the time interval between the actinic and the probe the sample must be in the vibrational coherent state).

These are shown in Figure 4.2 and .

In all the diagrams the first two interactions involve the pump, while the last two interactions involve the probe. The time ordering between pulses, together with the conservation of energy and the phase-matching condition let survive only eight diagrams. Let us take in consideration odd diagrams in the figures, in the off-resonant limit they are all equal, summed each other.

The first diagram in Figure 4.2 and the third have a signal function: S1(ω) = 2µ2 baµ 2 bc ~2 = Z ∞ −∞ dω1dω2Ea(ω1)Ea∗(ω2)Ep(ω − ω1+ ω2)Ep∗(ω)

(ω1− ωba+ iΓba) (ω1− ω2− ωca+ iΓca) (ω − ωba+ iΓba)

S3(ω) = 2µ2 baµ2bc ~2 = Z ∞ −∞ dω1dω2Ea∗(ω1)Ea(ω2)Ep(ω + ω1− ω2)Ep∗(ω)

(−ω1− ωba+ iΓba) (−ω1+ ω2− ωca+ iΓca) (ω − ωba+ iΓba)

Imposing the off resonant limit:

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4.1. NON RWA DIAGRAMS 47

Figure 4.2: Diagrams computed using the SRW A(eqn. 3.18). The last two interactions

always respect RWA, the first two diagrams compleately respect the RWA and are the only two diagrams that describe the signal for resonant IVS in the ground state. The other two diagrams have the first two interactions that both violate the RWA.

Figure 4.3: Diagrams computed using the SN O RW A(eqn. 3.19). The two last

inter-actions never respect the RWA approximation; in the last two diagrams every arrow does not respect RWA, but the energy is conserved at the end of the process.

Deciphering the non resonant response in impulsive raman spectroscopy

Sapienza – Universit`

a di Roma

FACOLT `

A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea in Fisica

Deciphering the non resonant response

in Impulsive Raman Spectroscopy

Lorenzo Monacelli

Prof. Tullio Scopigno

Contents

Chapter 1

Introduction

Chapter 2

Ultrafast spectroscopy

2.1

Transient Absorption

2.2

Ultrafast vibrational experiments

2.2.1

Impulsive Spontaneous Raman

2.2.2

Femtosecond Stimulated Raman Spectroscopy

2.2.3

Impulsive Vibrational Spectroscopy

Chapter 3

Quantum theory of

nonlinear spectroscopy

3.1

The interaction Hamiltonian

3.1.1

Second quantization of the atomic Hamiltonian and

Electric field

3.1.2

Optical response function

3.2

Time evolution of the density matrix

3.2.1

Interaction picture in the Liouville space

3.2.2

Non interactive evolution in the Liouvile space

3.2.3

Empirical description of the dephasing.

3.3

Static absorption

3.4

Feynman rules in time domain

3.4.1

Feynman rules in Fourier Space

3.5

Rotating wave approximation

3.6

Phase matching

3.7

Third order process: Transient Absorption

3.8

Raman scattering

Chapter 4

Modelling Impulsive

Vibrational Spectroscopy

4.1

Non RWA diagrams