Electron hole signatures in two-dimensional Stark spectroscopy

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Electron hole signatures in

two-dimensional Stark spectroscopy

Bachelor Thesis in Physics Author: Carlo Schimmel, s1908499 First supervisor: Thomas La Cour Jansen Second supervisor: Maxim Pschenitchnikov

July 7^th, 2016

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Introduction

In this thesis, we will construct a simple model that incorporates a linear Stark shift in the calculation of a two dimensional spectrum in order to give an indication of the possibilities that adding a Stark shift to two-dimensional spectroscopic techniques may have. First we will describe the general principles of two-dimensional spectroscopy and Stark spectroscopy in this chapter. We then proceed by developping a simple model that can be used for the simulation of two dimensional spectra and include a Stark shift into this model. Then in chapter three we present the results of simulations of this model for simple abstract systems, in order to evaluate the potential of combining the experimental techniques of two-dimensional and Stark spectroscopy in the conclusion in chapter 4. Finally, we discus the used method and suggest further extensions of the model in chapter 5.

1.1 Two-dimensional spectroscopy

Two-dimensional spectroscopy is a relatively recently developped technique that can be used to examine the electronic properties of complex molecules. As such, it can be used to analyze such things as peptides, proteins, DNA, biological photosynthetic systems and materials that can be used in the development of solar cells. [1] This is done by using several short laser pulses to probe to structure of a given system, usually with wavelengths in the infrared range so that this technique is also refered to as Two Dimensional Infrared Spectroscopy (2DIR), although recent developments have been made into using light pulses with wavelengths in the ultraviolet and visible light range as well. [1] The basic principle is an extention of more simple linear optical spectroscopy, in which the linear absorption spectrum of a system is obtained by analyzing the relative intensity of light of various wavelengths when it is directed through the system. Photons matching the energy gap between possible states will be absorbed and cause electronic excitations in the system, reducing the intensity of light passing through the system at the according wavelength.

In two-dimensional spectroscopy, the system under study is perturbed by 3 short laser pulses separated in time. The optical response is then a function of the time intervals between these interactions, which depends on the internal structure of the system and can thus be used to analyze this structure, such as the strengths of the couplings between molecular vibrations. [2] [3] Although the signal as a function of time is hard to visualize and intuitively unclear, it can be made highly informative by using a spectrometer, which efectively takes the Fourier transform of the singal, converting the response function from the time domain to the frequency domain with respect to the coherence times. These are the times between the first and second interaction and the time after the final interaciton for which the density matrix describing the system is a coherence of pure states. During the time between the second and final interaction, the system is in a population state and this time is treated as a parameter in the two-dimensional transformed response function. [2] [3] This function is then typically plotted as a contour plot for various waiting times. A schematic illustration of a general experimental setup for obtaining a two-dimensional spectrum is given in the figure on the next page:

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Figure 1.1: Experimental design for obtaining 2D spectra. Image taken from [2] (a) is an illustration of a direct measurement of the signal in the time domian, in (b) a spectrometer is included to measure the frequency response In order to understand how the appearance of a given two-dimensional spectrum gives information on the structure and properties of the sample under study, many computational techniques for simulating the effects of various contributions to the two-dimensional spectra for simple and more complicated systems have been developped in order to gain an understanding of the underlying structures that determine the outcome of 2DIR experiments. [1]

[4] [5] This thesis follows this line of research, as it will develop a simple model to examine how the presence of an external electric field will influence the appearance of a two dimensional spectrum. In chapter 2, we will present the analytical tools necessary for modelling and simulating two-dimensional spectra.

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1.2 Stark spectroscopy

Like linear and two-dimensional spectroscopy, Stark spectroscopy is an experimental technique that can be used to probe the electronic structure of complex molecules. This technique is a modification of linear absorption spectroscopy based on the Stark effect, which is the splitting of electronic states when the sample is placed in an external electric field. [6][7] It is also often called electrochromism or electro-optic absorption, because the effect of the field on the absorption and emmission spectra of a material is that the spectral lines are shifted. This shift is proportional to the changes in dipole moment ∆µ and polarizability ∆α upon excitation. A schematic illustration of the effect is given in the figure below.

Figure 1.2: Schematic illustration of the Stark effect on a linear absorption spectrum. Panel a illustrates the interaction for two populations with ∆µ alligned with or against the field. Panel b gives the corresponding absorption spectra with (dashed) and without (solid) the presence of the electric field. The Stark spectrum is then typically represented as the difference between these two spectra. (Field on - field off) The image was taken from [7].

Figure 1.3: Schematic illustration of a typical sample cell used for measuring Stark spectra. Image taken from [7]

Stark Spectrosocopy can be used to measure the changes in dipole moment and polarizability by using a controlled external field of known strength. These parameters can then be deduced by considering the first two derivatives of the lineshape of the Stark spectrum. [7][8] This is particularly useful since the change in dipole moment upon excitation gives information on the amount of charge transfer within a molecule that occurs when it is excited.

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Figure 1.4: Example of absorption and Stark spectra of spinach photosystem II reaction center, simultaneously recorded at 77 K with an external electric field strenght of 0.225MV/cm⁻¹. Image taken from [8]

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Chapter 2

Theory

In this chapter we will outline the main theory used in the development of our model in chapter 3. We begin by developing some basic terminology and notation in section 1, which we will use extensively further on. We then proceed to give a theoretical description of linear absorption spectra in section 2, which mainly serves as a stepping stone in order to give a detailed description of the computation of two-dimensional spectra in section 3. Then in section 4, we introduce a theoretical framework for linear Stark spectroscopy. In the final section of this chapter we specify the unit system that we will use for our calculations. Using all this, we will develop a model that incorporates a Stark shift into the computation of 2D spectra in the next chapter.

2.1 Formalism

In this thesis we will use a simple excitation Hamiltonian for a collection of N interacting electronic transitions with a single excited state, coupled to a classical oscillating bath coordinate.

H =

N

X

n=1

nB_n^†Bn+ X

n6=m

Jn,mB_n^†Bm (2.1)

Here, _n is the energy of transition n, J_n,m is the strenght of the coupling between transitions n and m, and B_n^† and B_n are the Bose creation and annihilation operators respectively. The time evolution of the wave function Ψ(t) of the system with the excitation Hamiltonian given by equation (2.1) is then given by the Schr¨odinger equation:

d

dtΨ(t) = −i

~

HΨ(t) (2.2)

We can relate the wave function at a time t0 to the wave function at some other time t by using a time evolution operator which is defined by:

ψ(t) = U (t, t0)ψ(t0) (2.3)

For a time independent Hamiltonian, the time evolution operator is thus given by:

U (t, t0) = exp

−i

~H(t − t0)

(2.4)

From this, we can easily derive some important properties of the time evolution operator that will be useful later on:

U (t₀, t₀) = exp{0} = 1 (2.5)

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U (t2, t0) = exp

−i

~

H(t2− t1+ t1− t0)

= U (t2, t1)U (t1, t0) (2.6)

U^†(t1, t0) = exp

+i

~H(t − t0)

= U (t0, t1) (2.7)

U (t₂, t₁) = U (t₂, t₀)U (t₀, t₁) = U (t₂, t₀)U^†(t₁, t₀) (2.8)

For simplicity, we will use a model with only two interacting excitations, which can be represented in a space of 4 states: the ground state |gi, which we use as the zero point of the excitation energy, the singly excited states |e1i and |e2i and a state where both sites are excited, |f i. A schematic representation of the model is given in the figure below:

Figure 2.1: Schematic representation of the model

Although we will not consider their precise physical origins in this thesis, the J couplings can, in their simplest form, be moddeled as dipole couplings which depends on the internal seperation of the different excitations within the molecule under study. Their strenghts can then be approximated by: [2]

Jij= 1 4π0

"

~ u_i· ~u_j

r³_ij − 3( ~r_ij· ~u_i)( ~r_ij· ~u_j) r_ij⁵

#

(2.9)

However, since we do not consider real systems, we will simply treat the coupling constant J as exogeneous parameters that can freely be changed. Then, if we use a simple Euclidian site basis given by:

|gi =





 1 0 0 0







|e1i =





 0 1 0 0







|e2i =





 0 0 1 0







|f i =





 0 0 0 1







The matrix representation of the Hamiltonian for a system of 2 coupled electronic transitions is given by:







0 0 0 0

0 ₁ J 0

0 J ₂ 0

0 0 0 ₁+ ₂







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For the development of the calculation of the linear and two dimensional absorption spectra, it is useful to represent the state of the system as a density matrix, which allows us to describe both populations of pure states and coherences between states, and also allows a discription in terms of statistical averages. For a pure state |ψi, the density matrix is given by:

ρ = |ψi hψ| (2.10)

For real systems however, we generally deal with statistical averages of pure states. This can not be incorporated in the description of the system as a wavefunction, but we can use the density matrix to describe the probability that the system is in a given possible pure state by defining:

ρ =X

k

Pk|ψki hψk| (2.11)

This description in terms of a density matrix is useful for an intuitive development of the linear and two dimensional response fucntions, since these are obtained by letting the density matrix interact with the transition dipole operator at the times of interaction with the light pulses that cause excitations, and then using the time evolution operator to propagate the density matrix in the interstate coherences to the times of the other interactions.

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2.2 Linear response

Before we turn our attention to the two-dimensional response of the system, we consider the linear response in order to gain a better understanding of the computations involved. The linear response function is obtained by considering the interaction of the system with two light pulses. We approximate these in the semi-impulsive limit [3], by treating the light pulses as short white light δ pulses at the specified interaction times. First, at time t = 0, we use a pulse to excite the system. If the energy of this pulse matches the energy gap between the ground state and excited state, a photon will be absorbed and cause a transition to an excited state. Then after some time t, we apply a second pulse, causing the stimulated emmission of a photon whose energy matches the energy difference between the ground state and the excited state of the electronic transition. The Feynman diagram describing this response for a single electronic transition is then:

Figure 2.2: Feynman diagram for the linear response

Then the intensity of the outgoing singal is obtained by multiplying the time evolution operator between times 0 and t with the transition dipole operator at time µ(0) from the right and µ(t) from the left. The transition dipole operator couples ground state of the system to the excited states. If we assume that the individual transition dipoles are constant in time, the times merely serve the indicate the times of the interaction. Thus, if the individual transition dipoles are µ₁and µ₂, the transition dipole operator is given by:

µ = µ₁(|gi he1| + |e1i hg| + |e2i hf | + |f i he2|) + µ₂(|gi he2| + |e2i hg| + |e1i hf | + |f i he1|) (2.12)

This is the full matrix dipole operator for the complete system. For the linear response, we only use the first row of this matrix, coupling the ground state to the singly excited states, and the first column, coupling the singly excited states to the ground state:

µ^ge= 0 µ1 µ2 0

µ^eg = (µ^ge)⁰ (2.13)

In terms of the density matrix, we assume that the system starts out in the ground state, which is described as a population:

ρ(t = 0) = |gi hg| (2.14)

At the time of the first interaction, the density matrix is acted upon the transition dipole operator to obtain an

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interstate coherence |ei hg|, and is then propagated between the interaction times using the time evolution operator for both sides of the denisty matrix. The linear absorption spectrum is than obtained as the correlation function of the transition dipoles at the interaction times through the time evolution operator. This is converted to the frequency domain by using a Fourier transform to give as an expression for the linear absorption spectrum: [5]

I(ω) = Im(

Z ∞ 0

i

~µ^geU (t, 0)µ^egΓ(∆t) exp{−iωt}dt) (2.15) In this expression we use the relaxation factor Γ(∆t) to account for the lifetime of the excitations in an ad hoc way, by letting the excitations decay exponentially with an excitation lifetime T . Its argument ∆t is the time difference between the first and second interaction. Since we have set t = 0 for the first interaction, this is simply the time variable t in our case. This causes peaks in the spectrum to smoothen out around the frequencies that correspond to the energies of the individual excitations. This relaxation factor is given by:

Γ(∆t) = Γ(t) = exp{−t/2T } (2.16)

Finally, we note that we have taken the Fourier transform in order to convert the signal in the time domain to the angular frequency domain. The angular frequency of the transition with energy gap is given through the relation:

= hc

λ = hν = ~ω (2.17)

Here h is Planck’s constant, c is the speed of light, λ is the photon wavelength corresponding to the energy gap, ν is the frequency, ~ = h/2π is the reduced Planck’s constant and ω is the angular frequency.

Thus, for a system with two transitions with energy gaps corresponding to the frequencies ω₁and ω₂and excitation lifetimes T , the linear spectra will have the following appearance:

Figure 2.3: Example of a linear spectrum

The positions of the peaks correspond to the frequencies of the transition, while the peak width depends on the reciprocal of the excitation lifetime T . The intensity of the peaks if proportional to their individual transition dipoles µ1 and µ2.

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2.3 Two-dimensional spectra

As in the case of linear spectroscopy, two-dimensional spectra are obtained by letting the system interact with three short laser light pulses and then measuring the response at some time after theses interactions. [2] This time we let the quantum system interact with short delta shaped laser pulses at times τ1, τ2, τ3and then measure the response at time τ4. The intervals between the interaction times are then denoted as t1, t2and t3respectively. This sequency of interactions gives rise to 6 possible Liouville pathways, which are schematically represented in the double sided Feynman diagrams below:

Figure 2.4: Liouville pathways for the 2D response. Image was taken from [1]

These pathways are referd to as Ground State Bleach (GB), Stimulated Emission (SE) and Excited state absorption (EA), and all three come in a rephasing and nonrephasing variety, depending on whether the first interaction is from the rigth (rephasing) or left (nonrephasing). If we denote the wave vectors of the applied laser pulses as ~k1, ~k2 and

~k3, we measure the signal emitted in the ~kI = −~k1+~k2+~k3direction for the rephasing signal and ~kII = ~k1−~k2+~k3

for the non rephasing signal. [2] [3] Whereas we only considered singly excited state to obtain the linear spectrum, we can now also have doubly excited states during t₃ in the excited state absorption pathway. As we did for the linear response, we include the lifetime of the doubly excited states in an ad hoc way, by assigning them a lifetime T₂. The lifetime of the singly excited states is now denoted as T₁. Futhermore, the total time interval under considera- tion is now the sum of the 3 time intervals between the interactions. Thus we now use the relaxtion factors given by:

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ΓGB(t3, t2, t1) = ΓSE(t3, t2, t1) = exp{−(t3+ 2t2+ t1)/2T1} (2.18)

ΓEA(t3, t2, t1) = exp{−(t3+ 2t2+ t1)/2T1− t3/2T2} (2.19)

We note here that the excited state absorption pathways are known to give rise to complications due to the appearance of peaks of opposite sign, as the corresponding pathways contain odd numbers of interactions from both sides as can be seen in figure 2.4. [4]. We will include these pathways in our calculation, but we will use the same relaxation factor as for the ground state bleach and stimulated emmission pathways.

ΓEA(t3, t2, t1) = ΓGB(t3, t2, t1) = ΓSE(t3, t2, t1) = exp{−(t3+ 2t2+ t1)/2T1} (2.20)

To calculate the two dimensional response, we consider each of these pathways separately in the same way as we did for the linear response, using the appropriate interactions of the transition dipole and time evolution operators with the density matrix for each pathway, and then sum over all three pathways for both the non-rephasing and rephasing signals. The individual contributions to the total two dimensional response as functions of the time intervals t₁, t₂and t₃for the 6 pathways are given by: [5]

S_GB^I (t3, t2, t1) = −(i

~)³[µ^ge(τ1)U (τ1, τ2)µ^eg(τ2)] × [µ^ge(τ4)U (τ4, τ3)µ^eg(τ3)]ΓGB(t3, t2, t1)

S_SE^I (t3, t2, t1) = −(i

~)³[µ^ge(τ1)U (τ1, τ3)µ^eg(τ3)] × [µ^ge(τ4)U (τ4, τ2)µ^eg(τ2)]ΓSE(t3, t2, t1)

S_EA^I (t3, t2, t1) = (i

~

)³[µ^ge(τ1)U (τ1, τ4)] × [µ^ef(τ4)U (τ4, τ3)µ^{f e}(τ3)] × [U (τ3, τ2)µ^eg(τ2)]ΓEA(t3, t2, t1)

S_GB^II (t3, t2, t1) = −(i

~

)³[µ^ge(τ4)U (τ4, τ3)µ^eg(τ3)] × [µ^ge(τ2)U (τ2, τ1)µ^eg(τ1)]ΓGB(t3, t2, t1)

S_SE^II (t3, t2, t1) = −(i

~

)³[µ^ge(τ2)U (τ2, τ3)µ^eg(τ3)] × [µ^ge(τ4)U (τ4, τ1)µ^eg(τ1)]ΓSE(t3, t2, t1)

S_EA^II (t₃, t₂, t₁) = (i

~

)³[µ^ge(τ₂)U (τ₂, τ₄)] × [µ^ef(τ₄)U (τ₄, τ₃)µ^{f e}(τ₃)] × [U (τ₃, τ₁)µ^eg(τ₁)]Γ_EA(t₃, t₂, t₁)

Here we use the full transition dipole operator µ (equation 2.9) for µêf and its transpose µ^{f e}= µêf⁰ in the calculation of the contributions for the excited state absorption pathways, while µ^geand µêg are as given in equation 2.10.

To obtain the total two dimensional response in the frequency domain, we sum the contributions for the rephasing and non-rephasing pathways separately and then take the Fourier transform with respect to times t1 and t3 for both. Finally, we sum these two contributions together to obtain the total two dimensional response as a function of the angular frequencies ω₁ and ω₃, which is parametrically dependend on the waiting time (also called mixing time) t₂. Note that we flip the sign of the first interaction for the rephasing response, since the first interaction in all these pathways is from the left.

S^I(ω3, t2, ω1) =

Z ∞Z ∞

[S_GBÎ (t3, t2, t1) + S_SEÎ (t3, t2, t1) + S_EAÎ (t3, t2, t1)] exp{i(ω3t3− ω1t1)}dt3dt1 (2.21)

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S^II(ω3, t2, ω1) = Z ∞

0

Z ∞ 0

[S_GBÎI (t3, t2, t1) + S_SEÎI (t3, t2, t1) + S_EAÎI (t3, t2, t1)] exp{i(ω3t3+ ω1t1)}dt3dt1 (2.22)

Now if we wish to visualize the total response function S^I(ω3, t2, ω1) + S^II(ω3, t2, ω1), we have several options. Since this function will in general be complex valued, we can plot either the real or imaginairy parts, or the absolute value of the function. These three options are illustrated for a single two level excitation in the figure below:

Figure 2.5: Example of the normalized real (left), imaginary (middle) and absolute value (right) contour plots of the 2D response of a single electronic transition

We observe that the spectra obtained by plotting the real part contains both positive and negative peaks, while these are absent in the plot of the imaginary part. (and obviousy in the absolute value plot as well) This appearance of negative peaks is due to the contributions from the excited state absorption pathways. [2][4] Both the real and imaginary parts are used throughout the literature [1][4][5]. In this thesis, we will follow [5] and use contour plots of the imaginary part of the total response function.

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2.4 Stark effect

The change in the linear absorption spectrum of a sample that occurs when an external electric field is applied is known as the Stark effect. [7] The external electric field causes a shift in the peak positions of the linear absorption spectrum, while leaving the line shape unaltered. This shift occurs through a shift in the absorption frequency ω, which has two contributions, obtained by taking the first two terms of a Taylor series in the external electric field E The first contribution is a linear shift proportional to the difference in dipole moment between the ground and~ excited state, while the second term is quadratic in the change of the polarizability upon excitation. [6] Thus the change in the peak frequency is proportional to the strenght of the external field and is given by:

∆ν = − 1

hc(∆~µ · ~E +1 2

E · ∆α · ~~ E) (2.23)

For our purposes however, we will not consider a real system but instead use an aproximation in which we look at a system of two coupled electronic transitions with only two states. Furthermore, we take the simplest possible case in which all the relevant parameters are aligned parallel, so that we can treat them as scalars rather then as vectors. As such, we are interested in the effect of the the induced frequency shift on the two-dimensional spectra in general terms, rather than the precise magnitude of this shift for a given field, which depends on the internal properties and orientation of a given system. Thus, we will use a more crude, yet simpler approximation to the Stark shift by only considering the linear term in equation 2.23. This reduces the number of parameters in the final model, especially when we treat the transition dipole moments of the transitions as scalars, as is often done in the literature for the sake of simplicity [1][4][5], and also concern ourselves only with the maginitude of the external electric field as a way of varying the size of the change in transition frequency. Since Stark spectra generally only reveal a shift in the linear absorption spectrum, as illustrated in the figure below, this approximation is likely to serve our purposes of analyzing the effect of the shift in transition frequency of the two-dimensional spectra.

Figure 2.6: Example of a linear stark spectrum for a single transition. The blue line is the original linear response, the green line is the shifted response due to the external electric field. The red line is the stark spectrum, the difference between the spectra with the field on and off.

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2.5 Computational considerations and the unit system

We will use a unit system in which we set ~ = c = 1. Using this, the energy gap of a transition can be directly written in terms of the angular frequency ω using equation (2.17). Note that setting c = 1 gives identical spatial and temporal frequencies since ν = _λ^c now equals ˜ν = ¹_λ. Furthermore, since this implies that we use h = 2π, the equation for the Stark shift (2.23), keeping only the linear term, can be conveniently rewritten to:

2π∆ν := ∆ω = −∆~µE (2.24)

Expressing the energy in terms of the angular frequency is also useful because the Fourier transform that we use in the computation of the linear (equation 2.15) and two dimensional spectra (equations 2.21 and 2.22) converts the singal in the time domain to the angular frequency domain. We will use units of inverse centimeters (cm⁻¹) for both the transition energies ω and the coupling strength J . The (changes in) transition dipoles then have units of Coulomb times centimeters (C·cm) and the strength of the electric field is given in Volts per centimeter (V cm⁻¹).

Using this, we can find the appropriate time scale by noting that the Fourier transform defines ω from the time as ω = ^2π_t. In numerical computations, the maximum value of ω for which the spectrum is calculated is determined by the size of the timestep, which we will scale so that the individual site frequencies ω1 and ω2 will lie within the range of the computation. For instance, if we wish to calculate the response for frequencies up to ωmax = 1000 cm⁻¹, we need a time step with a value of about _1000cm^2π −1 = 0.00628 cm. Using the fact that we have set 1 = c = 2.9979 × 10¹⁰ cm/s, we find that this corresponds to a time step of about 0.209 picoseconds in SI units. The resolution is then determined by the number of time steps, i.e. if we use 100 timesteps of 0.00628 cm we are effectively calculating the response at intervals of 10 cm⁻¹. Because time needed for doing the calculations depends heavily on the number of timesteps used, we will choose relatively low values for the excitation energies ω1

and ω₂, so that we can obtain a clearer resolution for a lower amount of timesteps, without having to calculate the response function for a large number of coordinates where the response will be almost zero anyway.

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Chapter 3

Results

In this chapter we will implement a linear Stark effect in the calculation of the two-dimensional spectrum of a system of two coupled electronic transitions. This will help us to gain a first understanding of the effect of adding a Stark shift in the calculation of a two dimensional spectrum, while keeping computation times relatively fast.

Adding the linear Stark shift in terms of the angular frequency (equation 2.24) to the exciton Hamiltonian given in equation 2.1 we get as our system Hamiltonian:

H = (ω₁− ∆µ₁E)B₁^†B₁+ J B^†₁B₂+ (ω₂− ∆µ₂E)B₂^†B₂+ J B^†₂B₁ (3.1)

We use this Hamiltonian and the formalism developped in the previous chapter to calculate linear and two dimensional Stark spectra and analyse how these spectra depend on the various parameters involved. Computations are done using Matlab versions R2015b and R2016b.

3.1 Model parameters

We have now developped a relatively simple model for analyzing two-dimensional Stark spectra, yet it depends on 9 paramaters already. The first six are the original frequencies of the transitions ω1, ω2, the strenght of their coupling J , their changes in dipole moment upon excitation ∆µ1 and ∆µ2 and the strength of the external electric field E. Furthermore, in the calculation of the two-dimensional spectra, we require three other parameters, which are the magnitudes of the transition dipoles µ1 and µ2 and the waiting time t2. However, we can reduce the number of relevant parameters to be analyzed a bit by first considering their effects theoretically . First we note that the sizes of the transition dipoles only determine the relative intensity of the peaks as in the calculation of the linear spectra (2.2). Thus we can simply set one of them equal to one and use the other as a means to scale the relative intensities of the peaks. Further simplification can be obtained by noting that a similar argument holds for the linear Stark effect. The linear Stark shift terms −∆µ_nE simply shift the effective transition frequency proportional to the external electric field and the change in transition dipole upon excitation of the individual excitons. Since we can treat the magnitude of the external electric field as an exogeneous variable that can be changed freely in an actual experimental setup, the main variable of interest in considering the Stark effect is the relative sensitivity to the external electric field of the transition energies. Since the magnitude of the shift can be scaled with the external electric field, we will simply set one of these equal to 1 (C·cm) and then use the other as a scaling parameter.

This leaves us with the task of finding appropriate numerical values for the transition frequencies and their couplings, where again we are mainly interested in their relative proportions since the model can easily be scaled. For typical real photosynthetic systems, the transition energies are in the order of a few electron volts. This corresponds to photons wavelengths in the range of visible light, i.e. several hunderds of nanometers, giving spatial frequencies of tens of thousands of inverse centimeters. The largest J-couplings for such systems are in the range of 50 - 150 cm⁻¹.[1] Thus, the J-couplings (in cm⁻¹) should be about 1/100 of the transition energies (in cm⁻¹). The numerical values for the transition energies and J-couplings that we will use in this chapter will be scaled down so that we

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can calculate two-dimensional spectra at a higher resolution, while keeping the time needed for the computations manageable.

3.2 Linear Stark spectra

As a benchmark, we first calculate the linear absorption spectra and Stark spectra for the model with the following parameters: ω1 = 250 cm⁻¹, ω2 = 350 cm⁻¹, J = 3 cm⁻¹. Because the effect of the external electric field is that the effective transition frequencies are shifted, we choose the values of ω1 and ω2 to be relatively far appart from each other. We will then consider what happens when we apply an external electric field that moves them closer together. Therefore, we will choose ∆µ1= 0.3 C·cm and ∆µ2= 1 C·cm. In this way, the lower transition frequency is shifted less than the higher one so that we can move them together. µ1 and µ2 are both set to 1 C·cm so that both peaks will have approximately the same intensity. We use time increments of 0.01 cm and a total number of 20, 000 timesteps. The vibrational lifetime is set to T = 0.03 cm or 1 picosecond. The Stark shift is calculated for external electric fields with strengths of 50, 100 and 150 V·cm⁻¹.

Figure 3.1: Linear spectra. Blue lines are the original spectra with no electric field, green lines the spectra with an applied electric field of indicated strenght and the red lines are the stark spectra (field on - field off). All original and shifted spectra are normalized with respect to the maximum value of the original, unshifted linear spectrum.

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Figure 3.2: Continued from figure 3.1

We observe that the shifts in the peaks are as expected based on the values we have chosen for the changes in transition dipole moment and the strengths of the electric field. Also, in the original spectra we see that the intensity of the peak at the higher excitation frequency ω2 is slightly higher than that at the lower excitation frequency ω1. This effect is due to positive J-coupling term. As the field is increased and the peaks move closer together, we see that the intensity of both peaks increases, which is most visible in the spectra for E = 100 V·cm⁻¹. When E = 150 V·cm⁻¹ the peak positions are shifted to ˜ω1 = 250 cm⁻¹− 0.3C·cm ×150 V/cm= 205 cm⁻¹ and

˜

ω2= 350 cm⁻¹− 1 C·cm ×150V/cm= 200 cm⁻¹. The peak position is at ω = 205 cm⁻¹ and the intensity is close to two times the intensity of the ω3peak in the original spectrum. Since the linear spectra are expected to be identical to the diagonal line in the two dimensional spectra, we expect to see similar effects when we calculate these in the next section.

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3.3 Two-dimensional Stark spectra

Using the same model parameters as in the last section, we now calculate the two-dimensional spectra with and without an external electric field of 50, 100 and 150 V·cm⁻¹. The only alteration that we make is that we reduce the number of time steps for both t1 and t3to 300 in order to obtain reasonable computation times. Further more, for all the spectra we calculate we will set the mixing time t2 equal to 0, as it does not influence the shape of the spectra in the model that we use, but merely decreases the total intensity, which only appears if we do not normalize the spectra. First, the two dimensional spectrum for the system, without the presence of an external electric field, is given below. The shifted and corresponding Stark spectra are given on the next page. As in the previous section, we normalize all the spectra with respect to the maximum value found in the unshifted two dimensional spectrum so that we can easily observe any changes in the intensity of the peaks.

Figure 3.3: 2D spectrum for ω1= 250 cm⁻¹, ω2= 350 cm⁻¹, J = 3 cm⁻¹, µ1 = µ2= 1 C·cm, t2= 0 In addition to the main peaks that we have seen as well in the linear spectrum, which coincides with the diagonal, we can now also directly see the presence of the coupling term J in the off-diagonal peak at (ω₁, ω₃) = (350, 250).

Again, due to the presence and positivity of this coupling, the intensity of upper peak is slightly higher than that of the lower one, despite the fact that we have set both their dipole moments equal to 1 C·cm. In the spectra given on the next page, we see that this shift of intensity from the lower to the higher peak becomes more evident as the strength of the external field is increased to bring the peaks closer together.

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Figure 3.4: 2D Shifted and Stark spectra

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We will now consider the effect of taking a smaller value for the transition dipole µ2, setting it to 0.3 C·cm.

Keeping the other transition dipole at 1 C·cm, we use this to represent a system where we have one optically dark state, such as a charge transfer state, in the near presence of a brighter exciton state. Furthermore, we will take

∆µ1= 0.3 C·cm and ∆µ2= 1 C·cm. In this way, we have an optically dark charge transfer state that is relatively insensitive to photo excitation, but highly senisitive to the external electric field, while the situation is reversed for the brighter exciton state. We can then use the external electric field to bring these states closer to another, which will cause the darker state to become more visible in the linear and two dimensional spectra as it is moved closer to the birghter state, which is due to the presence of the coupling. In order to make the effect itself more directly visible, we will also increase the value of the J-coupling to 25 cm⁻¹. This is a relatively large value, but it serves our purpose of making the increasing intensity of the charge transfer state with the lowered transition dipole more directly apparent in both the linear and two-dimensional spectra. All other parameters are kept the same. The linear Stark spectra for these parameters and various values of the electric field are given below. The two-dimensional Stark spectra are given on the next page.

Figure 3.5: Linear (Stark) spectra for ω₁= 250 cm⁻¹, ω₂= 350 cm⁻¹, J = 25 cm⁻¹, µ₁= 1 C·cm, µ₂= 0.3 C·cm,

∆µ₁= 0.3 C·cm, ∆µ₂= 1 C·cm, t₂= 0, T 1 = T 2 = 1 ps. Again blue lines are the original zero field spectra, green lines the shifted spectra and red lines the Stark spectra (field on - field off)

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Figure 3.6: Stark Spectra for ω1 = 250 cm⁻¹, ω2 = 350 cm⁻¹, J = 25 cm⁻¹, µ1 = 1 C·cm, µ2 = 0.3 C·cm,

∆µ₁= 0.3 C·cm, ∆µ₂= 1 C·cm, t₂= 0, T 1 = T 2 = 1 ps. All have been normalized with respect to the maximum of the unshifted 2D spectrum.

Figure 3.7: Continued from figure 3.6

Here we observe that the darker state can be made more clearly visible by applying a Stark shift. Initially, the peak corresponding to the transition with the smaller transition dipole is located at the positions indicated by 1, which is broadend and increases in intensity when it is shifted towards 2. The cross peak corresponding to the coupling shows similar behaviour which increases when the strength of the field is increased to bring the peaks closer together, as is seen from the original cross peaks indicated by a 3, that are shifted to the positions indicated by 4 and the increase in their intensity. The same effect is still visible in the Stark spectrum for E = 100 V/cm on the next page. Even when the peaks themselves overlap nearly perfectly when the strength of the field is set to E = 150 V/cm, a cross peak still appears in the two dimensional spectrum.

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Chapter 4

Conclusion

We have developped a simple model for the simulation of two-dimensional Stark spectra of an electronic dimer system. From the simulations of the two-dimensional Stark spectra, we have observed that the Stark effect can be used to shift both the diagonal and off-diagonal peaks together. In the presence of a positive coupling term J, this leads to an increase in the intensity of the upper peak. This effect is observed in both the linear and two-dimensional Stark spectra. In the Stark spectra, we also observe this effect of increasing intensity as the transition frequencies are moved together in the off diagonal peak corresponding to the couplings. In principle, this effect could thus be used to obtain information on electronic transitions that have a relatively small transition dipole moment compared to other nearby excitations. The main condition for this to be possible is that the changes in transition dipole moment ∆µ of the nearby excitations are unequal, so that we can decrease the differnces in their excited state energy levels by applying an external electric field of the appropriate strenght.

However, this effect can also be observed in the linear spectra presented in section 3.2. Nonetheless, the two- dimensional Stark spectra could still be more useful due to the appearance of the cross peaks. We note that in the linear spectra we observe that the two peaks begin to overlap at a certain point as the difference between the energy gaps is reduced. When this overlap occurs it becomes more difficult to distinguish between the peaks corresponding to the individual transitions, depending on the width of the peaks that is included in our model in an ad hoc way in the form of the vibrational lifetimes T . The same happens to the cross peaks as they move closer to the diagonal and start to interfere with the diagonal peaks, but their presence can still be observed in the inhomogenous broadening of the combined peaks, as is visible in the spectra for E = 100 V/cm in the original version of the model with a low value of the J coupling at the beginning of section 3.3, and even more so in the spectra with the increased J coupling and decreased transition dipole µ₂. Here, even when the peaks overlap and blur together in both the linear and two-dimensional spectrum for E = 150 V/cm, a clearly visible crosspeak still appears. Though it is beyond the scope of this thesis to establish whether or not the appearance of these crosspeaks truly gives us more information about the system under study than could be obtained from just the linear Stark spectra alone, we have found some indication that this might indeed be possible.

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Chapter 5

Discussion and outlook

The model we have used in this thesis is a very rough approximation for real systems and as such it only serves as a first indication of the possibilities that combining Stark Spectroscopy with two dimensional spectroscopy may have. One of the first steps in expanding this model would be to include vibrational couplings. The basics of this are explained in the next section. Including this also makes it possible to vary the waiting times t₂which can then be used to study the dynamics of the electron-hole formation after an initial excitation is created by the interaction with the laser pulses.

Another logical extention would be to consider the orientations and polarizability of the dipoles. Throughout this thesis we have assumed that these parameters are alligned parallel with respect to each other and the external electric field, which is generally not the case for real systems. Thus we could inlucidne the quadratic term in the Stark shift (equation 2.23) and treat all the parameters as vectors rather than as scalars. Another possible extention is to consider the effect that the Stark shift may have on the strength of the couplings between the excitations, which we will discuss briefly during the final section.

5.1 Vibrational coupling

We can include the effect of molecular vibrations into to the model by assigning to each unit a classical bath coordinate x that oscillates in time, which effects the transition energies ω through a coupling parameter λ. The Hamiltonian (2.1) is then extended to:

H =

N

X

n=1

(n+ λxn)B_n^†Bn+ X

n6=m

Jn,mB_n^†Bm (5.1)

The motion of the coordinate can then be modeled as a Brownian oscillator, treating it as a damped harmonic oscillator that is driven by a random thermal force. The dynamics of the coordinate are then given by the Langevin equation, which can be solves numerically using for instance the Euler method: [1]

m¨x = −kx − mγ ˙x + F^T (5.2)

Where the thermal force is taken from a normal distribution with standard deviation (2γkBT /∆t)^1/2. Here m is the mass of the oscillator, γ and k are the friction and spring constants for a classical damped harmonic oscillator, T is temperature and k_Bis Boltzmann’s constant. ∆t is the time step that we use to numerically solve this equation. For simplicity, we may assume that all parameters are the same for each coordinate, although we will assume that the stochastic behaviour of the coordinates is independent, i.e., we generate independent thermal fluctuations for each coordinate. Now because the coordinate x is fluctating in time, the excitation energy of each individual unit is also time dependent. The effective transition energy for site n is then a time dependent function ˜ωn(t) = ωn+ λxn(t), and therefore the Hamiltonian (equation 2.1) becomes parametrically time dependent when λ 6= 0. This implies that

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equation 2.4 is no longer valid. However, we can still use it as an approximation by assuming that the Hamiltonian is effectively constant during a small time interval ∆t and then calculating the time evolution operator from each interval to the next using the properties given in equations 2.5 - 2.8 so that we have [5]:

U (n∆t, 0) =

m=n

Y

m=1

U (m∆t, (m − 1)∆t) (5.3)

Where we would then use the same time step in the calculation of the time evolution operator as in the propagation of the classical coordinates. A schematic representation of the model coupled to a classical bath is given in the figure below:

Figure 5.1: Schematic representation of the model (equation 5.1) including a vibrational coupling. Image taken from [1]

The inclusion of the coupling of the electronic transition to molecular vibrations would however require us to average out our measurements over many simulations to properly include the vibrations, which would imply that the calculation times necessairy to obtain the spectra would drastically increase.

5.2 Stark effects on the electronic coupling

We briefly mentioned in section 2.1 that the J couplings can be modeled as dipole couplings using equation 2.9:

J_ij= 1 4π0

"

~ u_i· ~u_j

r³_ij − 3( ~r_ij· ~u_i)( ~r_ij· ~u_j) r_ij⁵

#

(5.4)

In this thesis we have not considered the precise physical origins of the J couplings but merely used them as parameters. We have also througout this thesis assumed that the transition dipoles are time independent, whill this is generally not true. When exposing a system to an external electric field of a strength that is typically used in Stark spectrosocopy, which is of the order of about 1 MV/cm⁻¹, we can expect this to influence the strenght and

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orientations of the transition dipoles in the sample, and even the changes that they undergo as electrons are excited to higher states. Another logical next step would thus be to include a more realistic modeling of the J-couplings and to consider how this may be affected by the presence of an external electric field.

5.3 Acknowledgement

Finally, the author would like to thank T.L.C. Jansen for his aid and constructive feedback during the research and writing of this thesis and J. Ogilvie for sharing some of her experimental data obtained by two-dimensional Stark spectroscopy that was included in the presentation of this work.

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Chapter 6

Bibliography

[1] R. Tempelaar, C.P. van der Vegte, J. Knoester and T.L.C. Jansen, Journal of Chemical Physics 138, 164104 (2013)

[2] P. Hamm and M. Zanni, Concepts and Methods of 2D infrared spectroscopy, Cambridge University Press (2011) [3] P. Hamm, Principles of Nonlinear Optical Spectroscopy: A Practical Approach, lecture notes retrieved on June

2, 2016 from http://www.mitr.p.lodz.pl/evu/lectures/Hamm.pdf (2005)

[4] A. G. Dijkstra and Y. Tanimura, The Journal of Chemical Physics, 142, 212423 (2015)

[5] T.L.C Jansen and J. Knoester, The Journal of Physical Chemistry B, 110, 22910 22916, (2006)

[6] L. N. Silverman, D.B. Spry, S.G Boxer and M.D. Fayer, The Journal of Physical Chemistry A, 112, 10244-10249 (2008)

[7] G.U. Bublitz and S.G. Boxer, Annual Review of Physical Chemistry, 48 : 21342 (1997)

[8] E. Romereo, B.A. Diner, P.J. Nixon, W.J. Coleman, J.P. Dekker and R. van Grondelle, Biophysical Journal, 103(2):185-94 (2012)

Electron hole signatures in two-dimensional Stark spectroscopy